Introduction to Computational Fluid Dynamics - Numerics - 4 - Classic Solver Algorithms

00:54:15
https://www.youtube.com/watch?v=I1ZO_B4JtRo

概要

TLDRIn this lecture, Professor Steve Miller discusses advanced solver algorithms in computational fluid dynamics (CFD), particularly focusing on incompressible and compressible flow algorithms. Key topics include the artificial compressibility method to address oscillatory pressure patterns, the SIMPLE algorithm for pressure-linked equations, and the PISO method for implicit pressure correction. The lecture also covers preconditioning techniques for low-speed flows and the FDB method for high-speed flows, highlighting their significance in commercial CFD solvers. The importance of eigenvalues in determining information travel speed and the role of parameters in the FDB method are also discussed, along with the checkerboard problem in numerical solutions.

収穫

  • 📊 Understanding advanced CFD algorithms is crucial for accurate simulations.
  • 🔄 The artificial compressibility method helps eliminate pressure oscillations.
  • 🛠️ SIMPLE and PISO are key algorithms for solving incompressible and compressible flows.
  • ⚙️ Preconditioning techniques improve convergence in low-speed flows.
  • 🚀 FDB is effective for high-speed and hypersonic flow simulations.

タイムライン

  • 00:00:00 - 00:05:00

    Professor Steve Miller introduces classic solver algorithms in computational fluid dynamics (CFD), focusing on incompressible and compressible flow algorithms. Previous discussions included steady and unsteady simulations, time integration, and numerical methods.

  • 00:05:00 - 00:10:00

    The artificial compressibility method (ACM) is explained as a technique to solve incompressible flow fields, addressing oscillatory pressure patterns in numerical solutions. The ACM modifies the continuity equation to include artificial compressibility terms that vanish at steady-state solutions.

  • 00:10:00 - 00:15:00

    The governing equations for incompressible flows are presented in non-dimensional form, with a focus on the continuity equation modified by artificial compressibility. The introduction of an artificial density term aims to eliminate fictitious oscillations in pressure calculations.

  • 00:15:00 - 00:20:00

    The eigenvalue analysis of the ACM is discussed, highlighting how disturbances in flow travel at modified velocities due to the introduction of artificial compressibility. This adjustment helps in solving stiff systems of equations more effectively.

  • 00:20:00 - 00:25:00

    The SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm is introduced as a method to eliminate oscillations in pressure and velocity solutions. The staggered grid approach is explained, along with the predictor-corrector procedure used in the algorithm.

  • 00:25:00 - 00:30:00

    The SIMPLE algorithm's steps are outlined, including initial pressure guesses, momentum equation solutions, and pressure corrections. The iterative nature of the algorithm is emphasized, along with its application in commercial CFD codes.

  • 00:30:00 - 00:35:00

    The PISO (Pressure Implicit with Splitting of Operators) method is presented as an alternative to SIMPLE, allowing for larger time steps and improved stability without iterative procedures. The governing equations for PISO are discussed, including predictor and corrector steps.

  • 00:35:00 - 00:40:00

    Preconditioning techniques are introduced for low-speed and high-speed flows, addressing the stiffness of equations in these regimes. The importance of modifying eigenvalues to improve convergence in numerical solutions is highlighted.

  • 00:40:00 - 00:45:00

    The FD (Finite Difference) method is discussed for compressible flows, emphasizing the need for flow field-dependent variations to handle high-speed and low-speed regions effectively. The role of parameters in controlling numerical stability and accuracy is explained.

  • 00:45:00 - 00:54:15

    The lecture concludes with a summary of the discussed algorithms, emphasizing their applications in commercial CFD solvers and the importance of understanding the underlying principles for effective simulation of fluid dynamics.

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ビデオQ&A

  • What is the purpose of the artificial compressibility method?

    The artificial compressibility method is used to solve incompressible flow fields and eliminate oscillatory pressure patterns in numerical solutions.

  • What does SIMPLE stand for in CFD?

    SIMPLE stands for Semi-Implicit Method for Pressure-Linked Equations.

  • What is the main advantage of the PISO algorithm?

    The PISO algorithm allows for pressure corrections without iterative procedures, enabling larger time steps and improved stability.

  • What are preconditioning techniques used for in CFD?

    Preconditioning techniques are used to improve convergence in low-speed or creeping flows by modifying the stiffness of the system of equations.

  • What is the FDB method used for?

    The FDB method is used for handling high-speed supersonic and hypersonic flows, particularly in scenarios with shockwave interactions.

  • How does the SIMPLE algorithm work?

    The SIMPLE algorithm involves a predictor-corrector approach to iteratively solve for pressure and velocity corrections in incompressible flows.

  • What is the significance of eigenvalues in CFD algorithms?

    Eigenvalues determine the speed at which information travels through the flow, affecting the stability and convergence of numerical solutions.

  • What is the role of the parameters s1, s2, s3, and s4 in the FDB method?

    These parameters control the influence of convection and diffusion terms in the governing equations, adapting the scheme to the flow characteristics.

  • What is the checkerboard problem in CFD?

    The checkerboard problem refers to oscillatory patterns in pressure solutions that arise from numerical artifacts in incompressible flow simulations.

  • What is the focus of the next lecture in this series?

    The next lecture will focus on stability analysis for simpler CFD schemes and how convergence is measured.

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  • 00:00:00
    welcome back to introduction to
  • 00:00:01
    computational fluid dynamics I'm
  • 00:00:03
    Professor Steve Miller today we'll be
  • 00:00:05
    talking about the classic solver
  • 00:00:06
    algorithms which reside in many
  • 00:00:08
    commercial CFD solvers in the previous
  • 00:00:13
    class we discussed the idea of unsteady
  • 00:00:15
    or steady simulations we then looked at
  • 00:00:17
    time integration and implicit explicit
  • 00:00:19
    schemes and we of course looked at the
  • 00:00:21
    numerical methods that operate to create
  • 00:00:24
    these types of solutions today we'll be
  • 00:00:26
    talking a little bit more advanced
  • 00:00:27
    solvers for particular types of being
  • 00:00:30
    compressible and then compressible flows
  • 00:00:31
    in particular we will be talking about
  • 00:00:34
    the so called incompressible flow
  • 00:00:36
    algorithms ACM simple and piezo and then
  • 00:00:40
    we'll look at the predominantly used for
  • 00:00:42
    compressible flow algorithms piezo which
  • 00:00:45
    is a modification of the incompressible
  • 00:00:47
    piezo algorithm preconditioning
  • 00:00:48
    techniques and of course F dB
  • 00:00:51
    let's first look at the artificial
  • 00:00:52
    compressibility method this is basically
  • 00:00:55
    used to solve incompressible flow fields
  • 00:00:57
    which of course comes from the knee
  • 00:01:00
    which is a little bit misleading is of
  • 00:01:02
    course it's compressible these are
  • 00:01:04
    basically pressure based formulations
  • 00:01:06
    and are used for incompressible flows to
  • 00:01:08
    keep pressure fields from becoming
  • 00:01:10
    oscillatory what that means is that
  • 00:01:12
    you'll see in a particular flow a
  • 00:01:14
    checkerboard like pattern in the
  • 00:01:16
    pressure data from the solver when you
  • 00:01:19
    look at the solution which appears to be
  • 00:01:21
    converged
  • 00:01:22
    so these checkerboard patterns appear
  • 00:01:24
    for certain reasons which you'll see in
  • 00:01:26
    a few minutes the ACM method is used to
  • 00:01:29
    of course try to eliminate these
  • 00:01:30
    checkerboard patterns in the pressure
  • 00:01:32
    solution this formulation is based on
  • 00:01:36
    field variables often involving the
  • 00:01:38
    pressure velocity and temperature where
  • 00:01:40
    compressible flows who are of course
  • 00:01:41
    look at and examine densities Momentum's
  • 00:01:44
    and energies so the main difficulty in
  • 00:01:48
    these incompressible solutions is of
  • 00:01:49
    course the calculation of the pressure
  • 00:01:51
    and that's apparent in simpler methods
  • 00:01:54
    where you find like an oscillatory
  • 00:01:56
    pattern and the pressure which is
  • 00:01:57
    fictitious because of the numerical
  • 00:01:59
    method let's return to our
  • 00:02:02
    incompressible governing equations we've
  • 00:02:05
    written these in index or Einstein
  • 00:02:06
    notation and in non-dimensional form we
  • 00:02:09
    have continuity for steady flows
  • 00:02:13
    and and here the variables with their
  • 00:02:18
    non dimensionalization save the Infinity
  • 00:02:20
    al-saleem scale and of course new some
  • 00:02:24
    infinities at ambient viscosity now in
  • 00:02:28
    the artificial compressibility method
  • 00:02:30
    the continuity equation will be modified
  • 00:02:32
    to include artificial compressibility
  • 00:02:34
    terms these will vanish when
  • 00:02:36
    steady-state solutions are reached for
  • 00:02:38
    example partial Rho tilde partial T
  • 00:02:40
    tilde plus VII is zero so you notice we
  • 00:02:43
    not much line C's equations here we
  • 00:02:45
    write with a tilde as an artificial
  • 00:02:48
    density so this artificial density we
  • 00:02:51
    want to drive to zero for a particular
  • 00:02:53
    or incompressible flow you notice that
  • 00:02:57
    term is missing on the continuity
  • 00:02:58
    operator nonetheless we are trying to
  • 00:03:02
    equate this term to the product of the
  • 00:03:05
    artificial compressibility factor beta
  • 00:03:07
    so we've introduced a new factor beta in
  • 00:03:10
    our equations of motion which we call
  • 00:03:12
    artificial and it's just a coefficient
  • 00:03:14
    which we set and we try and drive this
  • 00:03:17
    term of course it's a small positive
  • 00:03:19
    coefficient but we try and drive these
  • 00:03:22
    terms so that this equation is going to
  • 00:03:24
    be true so let's write out our equations
  • 00:03:27
    in vector notation so it can be more
  • 00:03:29
    compact and we'll try and find an eigen
  • 00:03:31
    value of it in the first equation we
  • 00:03:33
    have of course our vector form or if the
  • 00:03:36
    definition is in W ad and B here w a WB
  • 00:03:41
    w this is all put and of course a vector
  • 00:03:44
    form for pressure and velocity so for
  • 00:03:46
    example in W we had a partial P partial
  • 00:03:48
    T and parcel BJ personalty right there
  • 00:03:51
    the coefficients of a and B are written
  • 00:03:54
    down here along with their particular
  • 00:03:56
    derivatives so you notice that these
  • 00:03:57
    coefficients are generally known in one
  • 00:04:01
    particular space and time and CFD
  • 00:04:03
    solution let's for an example look at
  • 00:04:06
    the eigenvalues of a rain here to find
  • 00:04:12
    eigenvalues we would take of course the
  • 00:04:15
    determinant of AI minus lambda I with
  • 00:04:17
    died any matrix or lambda I are the
  • 00:04:20
    eigenvalues for each particular order of
  • 00:04:24
    the matrix so if a is of course four by
  • 00:04:26
    four we should
  • 00:04:27
    for eigenvalues if we do the hard work
  • 00:04:30
    we can find that the eigenvalues are
  • 00:04:31
    going to be u u and u plus the square
  • 00:04:36
    root of U squared plus beta and u minus
  • 00:04:38
    the square root and u minus the square
  • 00:04:41
    root of U squared plus beta this is very
  • 00:04:44
    much like the eigenvalue analysis which
  • 00:04:45
    i talked about earlier in the class here
  • 00:04:48
    we call beta is now viewed as an
  • 00:04:51
    artificial speed of sound or artificial
  • 00:04:53
    compressibility of course if get beta
  • 00:04:55
    goes to zero then we return and find the
  • 00:04:57
    original form the caressable equations
  • 00:04:59
    the original eigenvalues this can
  • 00:05:02
    interpret it very physically in that you
  • 00:05:04
    can view that the disturbances in the
  • 00:05:07
    flow are traveling at a velocity U
  • 00:05:09
    velocity U and have law say u Plus u and
  • 00:05:13
    a velocity of u minus u so you can see
  • 00:05:17
    if u is near zero or perhaps the mean
  • 00:05:20
    flow velocity for compressible flow then
  • 00:05:22
    I'll have some eigenvalues which are
  • 00:05:24
    very very close to zero this is a
  • 00:05:26
    problem and it might maintain or create
  • 00:05:28
    a system which is rather stiff and that
  • 00:05:30
    is meaning it's slang in a sense that
  • 00:05:33
    it's hard to solve with a linear algebra
  • 00:05:35
    technique or a numerical technique in
  • 00:05:39
    general nonetheless by adding beta you
  • 00:05:41
    can see that we move the potential eigen
  • 00:05:43
    values for some particular problems away
  • 00:05:45
    from their zero components and this is
  • 00:05:48
    excellent and a very wise thing to do is
  • 00:05:51
    of course then the information will
  • 00:05:52
    travel the numeral data through the
  • 00:05:54
    solution added to domain quicker we
  • 00:05:57
    would like to keep beta high enough such
  • 00:05:58
    that the pressure waves will be allowed
  • 00:06:00
    to travel enough to balance the viscous
  • 00:06:02
    effects of the solutions which are
  • 00:06:04
    usually found which we're going to show
  • 00:06:06
    in this particular ACM method through
  • 00:06:08
    the typical crate Nicholson method note
  • 00:06:10
    that we showed in a slide in the
  • 00:06:12
    previous lecture the crank Nicholson
  • 00:06:14
    method now by doing this by having some
  • 00:06:18
    positive coefficient beta will now have
  • 00:06:20
    a well-defined coefficient or set of
  • 00:06:22
    equations so the ACM method can be
  • 00:06:25
    summarized in by simply adding in some
  • 00:06:27
    artificial speed of sound or
  • 00:06:28
    compressibility factor to the
  • 00:06:30
    incompressible equations of motion and
  • 00:06:32
    simply solving them we have a typical
  • 00:06:34
    well-known technique like crank
  • 00:06:36
    Nicholson by doing this we will have
  • 00:06:37
    removed this so-called checkerboard
  • 00:06:39
    pattern we can see in some
  • 00:06:41
    Solutions you might come across this
  • 00:06:43
    yourself one day in some commercial
  • 00:06:44
    solvers and you'll have to change the
  • 00:06:46
    solver method especially if you're an
  • 00:06:49
    incompressible flow now if we do not use
  • 00:06:52
    this particular scheme of artificial
  • 00:06:54
    compressibility then two other most
  • 00:06:56
    popular approaches are the so-called
  • 00:06:58
    simple and piezo approaches neither
  • 00:07:00
    these are really simple to understand or
  • 00:07:02
    implement but their acronyms are aren't
  • 00:07:07
    just kind of a nice one because of
  • 00:07:08
    course it turns out to be simple but
  • 00:07:10
    it's nothing but simple so you might be
  • 00:07:13
    wondering what simple stands for it's an
  • 00:07:14
    acronym and it's called which I'll call
  • 00:07:16
    it simple for now on to just save time
  • 00:07:19
    it'll be the simple implicit method for
  • 00:07:21
    pressure linked equations what a
  • 00:07:23
    mouthful this also can be used to try
  • 00:07:27
    and eliminate the so-called oscillations
  • 00:07:29
    of the solution which we call with like
  • 00:07:31
    a slang word of checkerboard patterns of
  • 00:07:33
    velocity R pressures in each direction
  • 00:07:35
    at every other node in the solution so
  • 00:07:37
    if you plot say a plane your solution
  • 00:07:39
    you see a checkerboard prop pattern is
  • 00:07:41
    it cannot be there it's actually a
  • 00:07:43
    numerical artifact and if you look at
  • 00:07:46
    the actual residual of the equations
  • 00:07:48
    you'll find out that the mass is
  • 00:07:49
    actually not being conserved this is a
  • 00:07:51
    problem too so you're not really finding
  • 00:07:53
    a numerical solution to your equations
  • 00:07:55
    of motion for incompressible flows and
  • 00:07:57
    you'll find that these particular
  • 00:07:58
    difficulties can be bypassed through the
  • 00:08:01
    use of so-called staggered grids within
  • 00:08:03
    the outer room those is simple what that
  • 00:08:05
    means is we might have a standard grid
  • 00:08:07
    which we create with our grid generation
  • 00:08:09
    package and then we're going to be using
  • 00:08:12
    the solver to only look at it from an
  • 00:08:14
    artificial sense as a stator grid for
  • 00:08:17
    example one set of the solver will look
  • 00:08:19
    at points 1 3 5 7 etc and the other part
  • 00:08:23
    of the solver will look at nodes 2 4 6 8
  • 00:08:27
    etcetera so one will look at even a
  • 00:08:28
    normal look at odd and if you go up to
  • 00:08:30
    the next row then we stagger it by an
  • 00:08:32
    increment of one we've already talked
  • 00:08:34
    about the predictor-corrector procedure
  • 00:08:36
    in the previous class here we'll look at
  • 00:08:38
    it again the I remember the idea in a
  • 00:08:40
    particular character procedure is to the
  • 00:08:42
    that we make an initial prediction which
  • 00:08:44
    is like a mid step and then we correct
  • 00:08:46
    it to the next time step or iteration of
  • 00:08:49
    the solver here we're going to do a
  • 00:08:51
    predictor corrector struck with the
  • 00:08:53
    so-called success
  • 00:08:54
    pressure corrector step of course that's
  • 00:08:57
    part of the name of the acronym and
  • 00:08:59
    we'll say that now we're going to
  • 00:09:00
    decompose the pressure which is the
  • 00:09:02
    pressure we really care about as an
  • 00:09:04
    estimated pressure with a bar and a
  • 00:09:06
    prime these are not the so-called
  • 00:09:08
    average pressures or fluctuating
  • 00:09:10
    pressures which we've already talked
  • 00:09:11
    about and which we'll talk about later
  • 00:09:12
    in turbulence modeling in the class
  • 00:09:15
    likewise we can do the same
  • 00:09:17
    decomposition that there's an estimated
  • 00:09:19
    velocities and corrected velocities
  • 00:09:22
    written with the bars and prime
  • 00:09:24
    respectively so we now have a type of
  • 00:09:26
    decomposition
  • 00:09:26
    nonetheless we'll link this in with our
  • 00:09:29
    particular solver into something like a
  • 00:09:31
    predictor correct or type method in the
  • 00:09:35
    simple method the semi implicit method
  • 00:09:37
    for pressure linked equations will try
  • 00:09:39
    and relate the pressure Corrections to
  • 00:09:41
    the velocity Corrections by
  • 00:09:42
    approximating momentum equations so now
  • 00:09:45
    that we have this decomposition we will
  • 00:09:46
    approximate them among equations as you
  • 00:09:49
    see here once again you can see now we
  • 00:09:51
    have a right-hand side of a correction
  • 00:09:53
    and I excuse me a yes correction and the
  • 00:09:58
    left-hand side will be the original
  • 00:09:59
    left-hand side momentum equations both a
  • 00:10:01
    nonlinear term you can also solve this
  • 00:10:03
    for u Prime like here on the right with
  • 00:10:05
    delta T terms now you'll notice
  • 00:10:07
    something interesting there's no viscous
  • 00:10:09
    terms in here that's of course because
  • 00:10:11
    it's not part of this part of the
  • 00:10:13
    approximation we've dropped the
  • 00:10:14
    nonlinear terms I'm excuse me the
  • 00:10:16
    viscous terms no we'll try and combine
  • 00:10:20
    these particular sets of equations with
  • 00:10:22
    the continuity equation so we decompose
  • 00:10:24
    the momentum equation with the corrector
  • 00:10:25
    step and now we're going to insert it
  • 00:10:27
    and combine it with of course the
  • 00:10:28
    continuity equation this is the
  • 00:10:30
    so-called and when we find the rather
  • 00:10:32
    popular pressure correction Poisson
  • 00:10:35
    equation named after of course the
  • 00:10:37
    mathematician but he didn't find this
  • 00:10:39
    equation it just has a similar form to
  • 00:10:41
    his of course famous PDE which also
  • 00:10:43
    carries his name so now we'll have a
  • 00:10:46
    correction of pressure with index I I
  • 00:10:49
    will go as the native density over the
  • 00:10:51
    delta T the time step times the change
  • 00:10:54
    of the axial divergence of velocity
  • 00:10:57
    minus the predicted divergence and
  • 00:11:00
    they'll be equal to this right hand side
  • 00:11:02
    upon simplification for I equals 1 and 2
  • 00:11:04
    this is a 2d example in formulation it
  • 00:11:07
    can easily be extended as
  • 00:11:08
    reading now we still have to enforce
  • 00:11:11
    explicitly mass conservation at the
  • 00:11:13
    current iteration before I go to the
  • 00:11:14
    next time step of course that is to make
  • 00:11:17
    the corrector step so we solve these
  • 00:11:20
    particular equations through the simple
  • 00:11:22
    decomposition of the predictor equation
  • 00:11:24
    method through the simple algorithm now
  • 00:11:26
    let's look at the actual form of the
  • 00:11:28
    simple algorithm I've made a photocopy
  • 00:11:32
    from a single part of Chung's book on
  • 00:11:35
    CFD which shows the method and of course
  • 00:11:39
    he drives it in much greater detail than
  • 00:11:40
    we have in this class but let's go
  • 00:11:42
    through the method together so that we
  • 00:11:43
    can understand it and keep them
  • 00:11:45
    equations in mind of the simple method
  • 00:11:47
    as we look at it step one or Part A will
  • 00:11:53
    guess a pressure P bar to each grid
  • 00:11:56
    point for the initial iteration we can
  • 00:11:57
    simply set it to an ambient value or
  • 00:12:00
    anything we want it's like the initial
  • 00:12:01
    condition then we'll solve the momentum
  • 00:12:03
    equations to find V bar
  • 00:12:05
    that's the predictor step at the
  • 00:12:07
    staggered grade I plus 1/2 I might as
  • 00:12:09
    one have j plus 1/2 and shave my as 1/2
  • 00:12:11
    so here they're not going to every other
  • 00:12:13
    grid point like we were showing or
  • 00:12:15
    discussing earlier we're actually going
  • 00:12:17
    to half grid points or points in between
  • 00:12:19
    this is just like a midpoint or method
  • 00:12:21
    which advances in time but now we're
  • 00:12:23
    doing the same operation to space so you
  • 00:12:26
    see this is the motivation for the
  • 00:12:27
    method it's exactly equivalent then I'll
  • 00:12:29
    solve the pressure correction so we've
  • 00:12:31
    made the prediction now we make the
  • 00:12:32
    pressure correction which we showed of
  • 00:12:34
    course for P Prime in the previous page
  • 00:12:36
    and since the corner grid points will be
  • 00:12:38
    avoided is we can't perform this
  • 00:12:40
    operation on the corners then the scheme
  • 00:12:43
    will be so-called semi implicit and not
  • 00:12:45
    fully implicit as we previously shown so
  • 00:12:48
    it's a semi implicit method the
  • 00:12:50
    correction of the pressure and the
  • 00:12:52
    velocity will then be performed with the
  • 00:12:54
    equations we've shown and here they are
  • 00:12:56
    of course again we'll find the pressure
  • 00:12:58
    at the new step as P U and be based on
  • 00:13:01
    of course the predicted values and of
  • 00:13:06
    course the corrected values so that's
  • 00:13:08
    step D a is shown here as a substitution
  • 00:13:12
    and notice the predictor step of course
  • 00:13:14
    contains the viscous terms that's very
  • 00:13:16
    important
  • 00:13:17
    now now that we've gone to the next time
  • 00:13:19
    level we can simply replace the previous
  • 00:13:22
    intermediate values of pressure and
  • 00:13:25
    velocity with the new corrector values
  • 00:13:26
    of P and B and we return to step B we
  • 00:13:29
    then iterate from b to e over and over
  • 00:13:33
    through every iteration so for example
  • 00:13:35
    if we've done 1,000 iterations step
  • 00:13:38
    iteration one will perform a through e
  • 00:13:40
    and iteration two through a thousand
  • 00:13:42
    perform e through b of course and repeat
  • 00:13:44
    and then every iteration can do a
  • 00:13:46
    convergence check this is a simple
  • 00:13:49
    implementation of the 2d incompressible
  • 00:13:51
    simple algorithm which can of course
  • 00:13:53
    excuse lis be extended to three
  • 00:13:54
    dimensions and there's many papers
  • 00:13:56
    written about this so when you select a
  • 00:13:58
    simple algorithm in the CFD code which
  • 00:14:00
    are usually commercially based to
  • 00:14:02
    solving incompressible flows this is the
  • 00:14:05
    algorithm that's actually solving the
  • 00:14:06
    equations we're looking at note this
  • 00:14:08
    does not contain a turbulence model and
  • 00:14:10
    these equations can become much more
  • 00:14:12
    complicated if of course we use this
  • 00:14:13
    simple algorithm with a particular
  • 00:14:15
    turbulence closure so make sure that you
  • 00:14:18
    understand that
  • 00:14:19
    very important point let's look at some
  • 00:14:23
    examples to illustrate graphically
  • 00:14:25
    what's happening with the simple method
  • 00:14:26
    the correction the pressure will
  • 00:14:29
    actually accelerate convergence if we
  • 00:14:31
    used as a standard integration technique
  • 00:14:33
    then integration that is the number of
  • 00:14:36
    iterations and computer wall time and
  • 00:14:38
    power will takes much longer compared to
  • 00:14:40
    a simple method so we can actually
  • 00:14:43
    modify the method slightly even more to
  • 00:14:45
    find even better values but of course
  • 00:14:48
    introducing some parameter alpha it's an
  • 00:14:51
    empirical factor so you see now the
  • 00:14:53
    pressure will be decomposed into its
  • 00:14:56
    predictor step in this corrector step
  • 00:14:58
    but the corrector step has an extra
  • 00:15:00
    coefficient alpha we just numerically
  • 00:15:02
    experiment and find a value of point 8
  • 00:15:04
    which seems to be optimal we don't have
  • 00:15:07
    of course the same two other terms or
  • 00:15:09
    nabla P and nabla P now really the
  • 00:15:15
    previous discussion is for
  • 00:15:17
    time-dependent flows where we have time
  • 00:15:19
    to moon in terms but this can become
  • 00:15:21
    even more efficient if we use and assume
  • 00:15:24
    that the flow is steady so we might use
  • 00:15:27
    the following finite volume
  • 00:15:29
    discretization
  • 00:15:30
    and you might recall that we just talked
  • 00:15:32
    about how we're looking at the grits
  • 00:15:34
    which are staggered and 1/2 steps and
  • 00:15:37
    you might recall that we are looking at
  • 00:15:39
    the particular values at half steps for
  • 00:15:41
    example I've 1/2 so for the control
  • 00:15:45
    volume view we might have a cell
  • 00:15:47
    centered value where my cursor is and we
  • 00:15:50
    might go into I 1/2 direction so the
  • 00:15:52
    control volume for you and of course P
  • 00:15:55
    are actually staggered relative to the
  • 00:15:57
    central control volume for P this is of
  • 00:16:00
    course part of the reason why they call
  • 00:16:02
    it a semi implicit method for pressure
  • 00:16:04
    linked equations it's not explicit it's
  • 00:16:06
    not implicit but it is semi implicit in
  • 00:16:08
    that it's a mixed scheme for study flows
  • 00:16:11
    the simple out room becomes even simpler
  • 00:16:14
    if you will and we can substitute a
  • 00:16:16
    conservation variable fee and alpha will
  • 00:16:19
    be called the under relaxation parameter
  • 00:16:21
    for subscripts P and n B so these P and
  • 00:16:25
    n B's will denote the node under
  • 00:16:27
    consideration within the two dimensional
  • 00:16:29
    finite volume grid now we can improve
  • 00:16:32
    convergence even more through a simple
  • 00:16:35
    little method we can write a sub e minus
  • 00:16:38
    sum of a n B of UE prime goes as AE of P
  • 00:16:43
    prime pima's PE prime then e will be
  • 00:16:46
    easy use of e prime plus te P prime P
  • 00:16:49
    minus PE prime this last point is simple
  • 00:16:52
    is not that important the big picture
  • 00:16:55
    what is important is the overall
  • 00:16:56
    algorithm you'll see there's certain
  • 00:16:59
    advantages this symbol but you'll also
  • 00:17:01
    note that it's an iterative procedure
  • 00:17:03
    and it's semi implicit what if we want
  • 00:17:06
    to try and improve the method a bit and
  • 00:17:07
    look at a truly implicit method which
  • 00:17:11
    might of course have better stability
  • 00:17:12
    characteristics during our Brussels
  • 00:17:14
    schemes
  • 00:17:14
    so another incompressible solver is
  • 00:17:16
    called piezo which is the pressure
  • 00:17:18
    implicit with splitting of operations
  • 00:17:20
    you'll see and have noted that simple
  • 00:17:25
    requires an iterative procedure so it
  • 00:17:28
    will obtain solutions that is PISA
  • 00:17:30
    without interest procedures and with
  • 00:17:33
    very large time steps without much
  • 00:17:34
    computational effort relatives a simple
  • 00:17:37
    scheme and other schemes this is why
  • 00:17:39
    it's one very popular scheme to select
  • 00:17:41
    in computational fluid dynamic
  • 00:17:44
    the conservation of mass will be
  • 00:17:46
    designed to satisfied with
  • 00:17:47
    predictor-corrector steps so is still of
  • 00:17:50
    course uses the same predictor-corrector
  • 00:17:51
    idea the governing equations will now be
  • 00:17:55
    a momentum equation and pressure
  • 00:17:57
    correction equation let's look at them
  • 00:17:59
    now so we have a left-hand side which is
  • 00:18:01
    discretized which is marches in time
  • 00:18:03
    with a velocity descritization between
  • 00:18:06
    the next time step and the current one
  • 00:18:08
    where we know the data and the right
  • 00:18:10
    hand side will be of course the stress
  • 00:18:12
    shear stress tensors and strain with the
  • 00:18:16
    pressure term say partial P partial X
  • 00:18:18
    and partial P partial Y at steps n plus
  • 00:18:21
    1 now note this is truly an implicit
  • 00:18:24
    scheme as we discussed before because of
  • 00:18:26
    course we have time values of n plus 1
  • 00:18:27
    remember n is the current time level and
  • 00:18:30
    n plus 1 is the time level which we're
  • 00:18:32
    trying to find and minus 1 would be a
  • 00:18:34
    time level in the past there'll be a
  • 00:18:36
    pressure correction associated with this
  • 00:18:38
    let's look at this the new pressure
  • 00:18:40
    which is discretized will go as a
  • 00:18:43
    negative 4 over delta T of the
  • 00:18:45
    differences in velocities minus this
  • 00:18:48
    viscous term so here s IJ of course is
  • 00:18:52
    the derivatives of some one convection
  • 00:18:54
    in viscous diffusion terms we've lump
  • 00:18:55
    them all together as a right-hand side
  • 00:18:58
    term and we can calculate that usually
  • 00:19:00
    because of course is the previous and
  • 00:19:01
    next time stuff it's a closed form
  • 00:19:03
    equation for of course didn't known and
  • 00:19:06
    unknown field variables at n n n plus 1
  • 00:19:08
    respectively here we listed the rest of
  • 00:19:12
    the equations and we define s IJ and of
  • 00:19:16
    course the shear stress no there's a
  • 00:19:18
    predictor step and then there's a to
  • 00:19:20
    corrector steps so the simple method
  • 00:19:22
    have one predictor step and two
  • 00:19:24
    corrector steps this one simply has to
  • 00:19:25
    you can take a second to look at the
  • 00:19:28
    form of equations in the current time
  • 00:19:29
    levels the first predictor step uses a
  • 00:19:32
    time level denoted by star or asterisk
  • 00:19:36
    superstar and the second one uses a
  • 00:19:38
    double supe star so through this simple
  • 00:19:41
    predictor and corrector algorithm we can
  • 00:19:43
    implement the Pizzo method for the
  • 00:19:46
    incompressible 2d system of equations
  • 00:19:49
    this is seems like a lot of work and
  • 00:19:52
    there are certain characteristics of
  • 00:19:53
    these schemes which are beneficial for
  • 00:19:55
    example their implicit which takes them
  • 00:19:57
    more power
  • 00:19:58
    from the next time stuff but the time
  • 00:19:59
    steps we can make our humongous and
  • 00:20:02
    maintain stability and we eliminate
  • 00:20:04
    those so-called oscillations of the grid
  • 00:20:06
    so we can greatly increase the stability
  • 00:20:08
    of the scheme by splitting s IJ term
  • 00:20:12
    which we defined here the nonlinear term
  • 00:20:15
    in shear stress term viscous term and
  • 00:20:18
    will split into his Dinoland 9 now terms
  • 00:20:20
    for example we would split s IJ with I
  • 00:20:25
    that is this tensor s IJ with
  • 00:20:28
    derivatives I as partial partial X of
  • 00:20:31
    Rho U feet minus K V partial X as an
  • 00:20:35
    example now we can expand this term
  • 00:20:39
    through simple discritization so here's
  • 00:20:42
    the continuum form one can discretize it
  • 00:20:43
    using of course our standard
  • 00:20:45
    differencing finite differencing
  • 00:20:47
    techniques which we talked about in our
  • 00:20:48
    numeric class and we can apply up
  • 00:20:51
    winding so up whining is going to take
  • 00:20:53
    these parts of the stencil at midpoints
  • 00:20:56
    and bias them towards the upwind
  • 00:20:58
    direction for example if the current
  • 00:21:01
    velocity is positive we would use this
  • 00:21:03
    so-called upwind direction and if it's
  • 00:21:05
    negative one so look at the differences
  • 00:21:08
    between the indices the positive
  • 00:21:11
    velocity you use I and I minus one and
  • 00:21:13
    the negative velocity we use indexes I
  • 00:21:14
    plus 1 and I so this means the stencil
  • 00:21:17
    is biased in the upwind direction that
  • 00:21:19
    uses more grid points or excuse me
  • 00:21:22
    nodes for the differencing approach in
  • 00:21:25
    the direction of the wind
  • 00:21:27
    so using these operations we can find
  • 00:21:30
    the P so with a splitting operation
  • 00:21:32
    which we just showed and we'll arrive at
  • 00:21:35
    these particular equations for Rho UV at
  • 00:21:37
    I plus 1 and I minus 1 with of course an
  • 00:21:41
    up winning operator which is equivalent
  • 00:21:43
    to of course positive and negative
  • 00:21:45
    velocities over its high speed you know
  • 00:21:49
    it might be a very very small negative
  • 00:21:51
    velocity a very high speed positive
  • 00:21:54
    velocity so we can also rewrite s IJ and
  • 00:21:57
    these coefficients alpha beta and gamma
  • 00:21:59
    and put them on the scheme once we do
  • 00:22:02
    all that we can try to analyze the
  • 00:22:04
    system for a structured grid and you'll
  • 00:22:06
    find a system like this a tri diagonal
  • 00:22:09
    dominant of beta gamma and alpha
  • 00:22:12
    beta and a gamma and alpha this is the
  • 00:22:16
    diamond off diagonal turn decomposition
  • 00:22:18
    which we talked about of course on the
  • 00:22:20
    first line of slide 15 now that's an
  • 00:22:24
    incompressible flow and the whole piezo
  • 00:22:27
    algorithm which you can of course
  • 00:22:28
    implement when you run the piezo
  • 00:22:31
    incompressible solver unsafe fluent or
  • 00:22:33
    star or other commercial software codes
  • 00:22:35
    you can imagine it's a lot of work to
  • 00:22:37
    implement these just for the laminar
  • 00:22:39
    equations but once again they can also
  • 00:22:41
    be implemented and applied to flows with
  • 00:22:44
    a turbulence model now what if we want
  • 00:22:47
    to consider a flow which has
  • 00:22:48
    compressibility which is perhaps in a
  • 00:22:50
    general sense seen as a flow with mach
  • 00:22:52
    number greater than 0.3 there's some
  • 00:22:55
    modifications in the algorithm which we
  • 00:22:56
    have to examine we would have to split
  • 00:22:59
    SI je once again to Dino Dino parts as
  • 00:23:02
    we discussed and we can use the piezo
  • 00:23:04
    algorithm apply to a compressible flow
  • 00:23:06
    we would once again have a predictor
  • 00:23:08
    step and to correction steps between the
  • 00:23:12
    correction steps we would solve for
  • 00:23:14
    pea-soup star minus PN which is known PN
  • 00:23:17
    is known this is the current time step
  • 00:23:19
    in the known value and certain to obtain
  • 00:23:21
    the new velocities V stubble soup star
  • 00:23:24
    those values would of course be used to
  • 00:23:27
    find and solve for Rho we can subtract
  • 00:23:30
    these two particular equations and find
  • 00:23:33
    this equation which is labeled from
  • 00:23:36
    Chung's book we would look for the
  • 00:23:39
    triple soup star which of course in this
  • 00:23:42
    case with I I would be to enforce the
  • 00:23:45
    conservation of mass we would then find
  • 00:23:48
    solutions which lead to this form where
  • 00:23:50
    VI
  • 00:23:51
    triple soup star and P Double soup star
  • 00:23:53
    is vo n plus 1 and and P n plus 1 so
  • 00:23:56
    this would complete the splitting
  • 00:23:58
    process where V triple star and piece
  • 00:24:00
    double soup star will imply the exact
  • 00:24:03
    solutions of V of n plus 1 P n plus 1
  • 00:24:05
    which can be shown right here in these
  • 00:24:07
    particular equations for these
  • 00:24:09
    particular procedures you can see the
  • 00:24:11
    full derivation and implementation in a
  • 00:24:14
    paper by Asuma Gausman and Watkins from
  • 00:24:17
    1986 this is the so-called PSO
  • 00:24:20
    compressible form
  • 00:24:22
    let's now examine the modification
  • 00:24:24
    explicitly for compressible flows we
  • 00:24:27
    would add an additional corrector stage
  • 00:24:29
    which would be incorporated because of
  • 00:24:31
    course now there's coupling between
  • 00:24:32
    momentum energy and pressure
  • 00:24:34
    there's fluctuations of density and
  • 00:24:36
    temperature with the effect of
  • 00:24:37
    compressibility which we did not have
  • 00:24:39
    before and they must be common form so
  • 00:24:41
    we can discretize a piezo scheme in a
  • 00:24:43
    particular time levels like I've written
  • 00:24:45
    down in the middle of the page you can
  • 00:24:47
    summarize a new predictor-corrector
  • 00:24:48
    steps as now we have a momentum
  • 00:24:52
    predictor step an energy predictor step
  • 00:24:55
    an intermediate momentum predictor step
  • 00:24:58
    and another momentum corrector step a
  • 00:25:01
    second one then we'll have an energy
  • 00:25:04
    corrector and finally a third momentum
  • 00:25:07
    corrector we can then and have to use a
  • 00:25:09
    continuity equation based on of course
  • 00:25:11
    the third corrector steps and the
  • 00:25:14
    original density and the pressure
  • 00:25:16
    equation which we can use to find a new
  • 00:25:20
    pressure corrector step equation these
  • 00:25:23
    are all the equations they're used to
  • 00:25:25
    find a two or three dimensional
  • 00:25:28
    compressible flow of the navier-stokes
  • 00:25:31
    equations with the pleasure implicit
  • 00:25:33
    splitting operation method so take a
  • 00:25:36
    second and just to review these
  • 00:25:37
    equations because of time in the class
  • 00:25:39
    you don't have to review these but if
  • 00:25:41
    you're curious and you're running piezo
  • 00:25:43
    methods you'll have to know that these
  • 00:25:44
    are what are programmed and validated
  • 00:25:47
    with most commercial solvers it's very
  • 00:25:49
    rare today to find research-based
  • 00:25:51
    solvers that use the PSO method we've
  • 00:25:54
    looked at incompressible flows and some
  • 00:25:57
    workhorse schemes implemented in most
  • 00:25:59
    commercial solvers to find the solutions
  • 00:26:02
    and we just looked at those equations
  • 00:26:03
    and gave you a very broad overview of
  • 00:26:06
    the algorithm to solve them now we'll
  • 00:26:08
    look at preconditioning techniques now
  • 00:26:10
    for very low speed flows which are
  • 00:26:12
    incompressible or creeping or very high
  • 00:26:16
    speed flows that are supersonic that
  • 00:26:18
    contain regions that are very very very
  • 00:26:20
    slow and locally incompressible then the
  • 00:26:24
    density based formulations which we
  • 00:26:25
    showed their chart traditional are very
  • 00:26:27
    slow to converge this is where the idea
  • 00:26:29
    of pre conditioning techniques came in
  • 00:26:31
    and you might have seen how we formed
  • 00:26:34
    coefficient matrices in the pre
  • 00:26:36
    previous methods of say for example
  • 00:26:38
    simple we might be able to use
  • 00:26:39
    preconditioning or linear algebra
  • 00:26:41
    techniques on those matrices to change
  • 00:26:44
    them from a very very stiff system of
  • 00:26:47
    equations to one which is less stiff or
  • 00:26:50
    one that can be handled by a numerical
  • 00:26:51
    solver with much better ease why are
  • 00:26:54
    very very slow speed flows difficult and
  • 00:26:57
    highly stiff to solve well that's
  • 00:26:59
    because the acoustic speeds are so much
  • 00:27:01
    higher than the flow velocity and the
  • 00:27:03
    mean velocity might be very very very
  • 00:27:05
    close to zero and this means that some
  • 00:27:08
    information is travelling very fast
  • 00:27:09
    through the fluid domain and someone's
  • 00:27:11
    very trimmed very very very slow and so
  • 00:27:14
    it would take a very long for the
  • 00:27:16
    information the Travelodge domain and
  • 00:27:17
    find say a steady solution for a flow
  • 00:27:20
    which is very very slow speed or very
  • 00:27:22
    high speed with a very slow speed region
  • 00:27:24
    so we'll try and examine the transition
  • 00:27:27
    from spatially varying compressible to
  • 00:27:28
    incompressible regions and vice-versa
  • 00:27:31
    this will use the density based
  • 00:27:33
    formulation and we'll use
  • 00:27:35
    preconditioning matrices on the
  • 00:27:38
    particular time-dependent terms this
  • 00:27:41
    will essentially and mathematically
  • 00:27:43
    improve the convection eigenvalues for
  • 00:27:45
    low Mach number or incompressible flow
  • 00:27:47
    regimes let's take a look at these
  • 00:27:49
    formulations now as before we'll rewrite
  • 00:27:52
    our equations in vector form which
  • 00:27:55
    Akopian so this is couldn't perhaps form
  • 00:27:57
    to a curvilinear or transform coordinate
  • 00:28:00
    system using a finite difference
  • 00:28:01
    approach we have a vector U which is our
  • 00:28:03
    solution vector of P the velocities and
  • 00:28:05
    temperature F and G of course our flux
  • 00:28:08
    vectors here will rewrite a as partial u
  • 00:28:13
    partial Q like we've done in the second
  • 00:28:15
    equation and Q will then be this
  • 00:28:18
    particular matrix so we take the partial
  • 00:28:20
    derivative of U with respect to Q and
  • 00:28:22
    then we marae write the first equation
  • 00:28:24
    as second with a equals partial u
  • 00:28:26
    partial Q here we have a alpha sub P and
  • 00:28:29
    beta sub T these coefficients are
  • 00:28:32
    defined at the lower part of the page so
  • 00:28:35
    it's a lot of work to do this type of
  • 00:28:36
    decomposition and rewriting the equation
  • 00:28:39
    in the form of the second equation on
  • 00:28:42
    the page nonetheless you can see is now
  • 00:28:44
    in a vector form which we've shown for
  • 00:28:46
    you now let's try and look at the eigen
  • 00:28:48
    values and the main
  • 00:28:49
    because the eigenvalues of this matrix
  • 00:28:51
    will dictate the types of speeds that
  • 00:28:54
    the information travels through the flow
  • 00:28:55
    as we're finding a steady solution so
  • 00:28:59
    let's look at the eigenvalues of a
  • 00:29:00
    through the same equation we did before
  • 00:29:02
    we'll take the inverse of a times B sub
  • 00:29:05
    I minus lambda I you can see here's a
  • 00:29:08
    and B matrices in the coefficient
  • 00:29:10
    equations you'll see that incompressible
  • 00:29:13
    limits that is the densities going to
  • 00:29:16
    some constant informally will find that
  • 00:29:18
    the eigenvalues become rather stiff
  • 00:29:20
    which is an algebraic set of equations
  • 00:29:23
    which are ll condition an accoustic
  • 00:29:25
    speeds will become infinite this means
  • 00:29:27
    that as the Mach number the flow becomes
  • 00:29:29
    very small and goes to zero and the
  • 00:29:32
    density becomes a constant and that
  • 00:29:33
    cannot fluctuate that is we're solving
  • 00:29:35
    the incompressible set of equations the
  • 00:29:37
    speed of sound will go to infinity this
  • 00:29:39
    is rather troubling this of course it's
  • 00:29:41
    the unphysical thing but is a property
  • 00:29:42
    of making the incompressible assumptions
  • 00:29:45
    that the acoustic waves travel at
  • 00:29:46
    infinity this is not appropriate for
  • 00:29:49
    finding an incompressible flow solutions
  • 00:29:51
    of course in true and compressible flows
  • 00:29:53
    acoustic waves do not in reality travel
  • 00:29:56
    at infinite speed they travel at roughly
  • 00:29:59
    the square root of DM a times R times T
  • 00:30:02
    the ratio specific heats times the gas
  • 00:30:04
    constant times the static temperature
  • 00:30:06
    and so we want to modify these
  • 00:30:09
    eigenvalues in some way and so that we
  • 00:30:12
    can make the system less stiff so we can
  • 00:30:14
    find it will do this through examination
  • 00:30:17
    the first column of the time to Cobian
  • 00:30:19
    which contains the derivative of the
  • 00:30:20
    density versus pect pressure at some
  • 00:30:22
    constant temperature so if we look at
  • 00:30:24
    partial Rho partial P at T it goes as
  • 00:30:26
    gamma over the speed of sound squared
  • 00:30:28
    this is a well-known equation in
  • 00:30:30
    compressible flow it relates the changes
  • 00:30:33
    of density with pressure at some
  • 00:30:35
    temperature and it'll go as gamma over
  • 00:30:37
    this beautys on squared so this
  • 00:30:40
    derivative will vanish for
  • 00:30:41
    incompressible flows in the speed of
  • 00:30:42
    sound go to infinity this is the core
  • 00:30:44
    problem written down mathematically to
  • 00:30:46
    get around this problem we're going to
  • 00:30:48
    modify density essentially by
  • 00:30:50
    multiplying it by beta
  • 00:30:52
    we'll have Rho a beta so T a partial Rho
  • 00:30:55
    partial P at some temperature and
  • 00:30:57
    expanding out the right-hand side we can
  • 00:30:59
    rewrite it as 1 over V R which is now a
  • 00:31:02
    reference velocity my
  • 00:31:04
    one over C sub P which is the
  • 00:31:05
    coefficient a specific heat at constant
  • 00:31:07
    pressure times density times partial Rho
  • 00:31:10
    partial T at some constant pressure you
  • 00:31:12
    can see the mathematical operations here
  • 00:31:14
    from thermodynamics to write up these
  • 00:31:16
    equations and expansions what might V
  • 00:31:19
    sub R be through this introduction we
  • 00:31:21
    might simply set it for the speed of
  • 00:31:23
    sound for compressible flows or the
  • 00:31:26
    square root of say the square of the
  • 00:31:29
    velocity for incompressible flows for
  • 00:31:32
    her house boundary layers as an example
  • 00:31:35
    so basically we do have to set it to
  • 00:31:37
    some reference velocity that shouldn't
  • 00:31:40
    be too much of a problem a signal
  • 00:31:42
    reference velocity for the flow we can
  • 00:31:43
    try to adjust the eigenvalues of the
  • 00:31:45
    pre-ignition system and their
  • 00:31:47
    adjustments will be done through of
  • 00:31:49
    course the eigenvalue definition with a
  • 00:31:51
    and we'll right now that capital lambda
  • 00:31:54
    goes to dyno of u u YY soup star plus
  • 00:31:58
    the speed of sound
  • 00:31:59
    - u superstar - the speed of sound soup
  • 00:32:02
    star we've now defined two new variables
  • 00:32:05
    Y sweep star YouTube star now capital
  • 00:32:08
    lambda will introduce as the diagonal as
  • 00:32:11
    which are the eigen values of the system
  • 00:32:14
    as u u u the swash should be u YouTube
  • 00:32:19
    star plus the speed of sound soup star a
  • 00:32:21
    soups are and YouTube star minus u soup
  • 00:32:25
    start so the three eigenvalues are you
  • 00:32:28
    you and you and the last two are you
  • 00:32:29
    plus the speed of sound and u minus B so
  • 00:32:33
    YouTube star and the speed of sound soup
  • 00:32:35
    star are now defined in this manner
  • 00:32:37
    these are found from of course the eigen
  • 00:32:39
    value operations with the substitution
  • 00:32:42
    of our modified speed of sound relation
  • 00:32:46
    you'll now find when the velocity is
  • 00:32:48
    greater than speed of sound that the
  • 00:32:50
    eigen values will become u plus the
  • 00:32:53
    speed of sound and u minus V the speed
  • 00:32:54
    of sound and if the reference velocity
  • 00:32:57
    is approximately zero are very very
  • 00:32:58
    small then all the eigen values will be
  • 00:33:00
    of the same order so if a very very
  • 00:33:02
    small creeping flow like here if the
  • 00:33:05
    overall velocity of the flow is say 0.01
  • 00:33:08
    meters per second then all the eigen
  • 00:33:10
    values will be modified to become of the
  • 00:33:12
    same order of U this is good this means
  • 00:33:14
    our system is not very Stephanie
  • 00:33:17
    easily solved with a numerical method by
  • 00:33:19
    simply applying this preconditioning
  • 00:33:20
    technique these preconditioning
  • 00:33:23
    techniques are actually pioneered in
  • 00:33:25
    linear algebra systems if we just have a
  • 00:33:27
    linear system of ax equals B we can
  • 00:33:29
    precondition the matrix a and B with the
  • 00:33:32
    same multiplication and find a
  • 00:33:35
    pre-condition system which modifies
  • 00:33:36
    Augen values and then we can solve the
  • 00:33:38
    new system of equations opposed to
  • 00:33:41
    solving the original set which might be
  • 00:33:42
    impossible this is exactly what we're
  • 00:33:44
    doing with this CFD technique we're
  • 00:33:46
    modifying the eigenvalues through a
  • 00:33:49
    modification of the speed of sound
  • 00:33:51
    relation with a reference value
  • 00:33:53
    introduction we can further improve the
  • 00:33:55
    efficiency in time accurate solutions we
  • 00:33:57
    may try and utilize a dual time stepping
  • 00:33:59
    method which we showed previously of
  • 00:34:01
    course for a midpoint type method for
  • 00:34:03
    simple and P so we'll introduce the
  • 00:34:05
    pseudo time derivative so for finding a
  • 00:34:07
    steady flow we will introduce a pseudo
  • 00:34:09
    time derivative and we'll try and drive
  • 00:34:11
    that to the zero in a linearized
  • 00:34:14
    iteration step form so we can now take
  • 00:34:16
    our vector equation which was shown on
  • 00:34:19
    slide 23 the second equation which we
  • 00:34:22
    discussed its formulation and we'll find
  • 00:34:25
    a right-hand side age which of course
  • 00:34:27
    will let be the compress the
  • 00:34:29
    incompressible form the continuity
  • 00:34:31
    equation so the pseudo time step here
  • 00:34:34
    DITA will approach infinity as the
  • 00:34:36
    pseudo time step vanishes and will
  • 00:34:38
    recover of course the steady original
  • 00:34:40
    governing equations that is this term
  • 00:34:42
    will go away over time and we have the
  • 00:34:44
    same terms from the modified vector
  • 00:34:47
    equation that's really the transition
  • 00:34:51
    from incompressible flow to compressible
  • 00:34:53
    and then looking at incompressible or
  • 00:34:55
    compressible flows of course with a
  • 00:34:57
    preconditioning technique let's turn our
  • 00:34:59
    attention to a completely different
  • 00:35:01
    concept for compressible flows that
  • 00:35:03
    contain both high speed flow and very
  • 00:35:05
    low speed flow flow field dependent
  • 00:35:08
    variation was developed to handle high
  • 00:35:10
    speed supersonic flows and hypersonic
  • 00:35:13
    flows where there's shockwave bound
  • 00:35:16
    chill interactions a shock wave boundary
  • 00:35:18
    layer interaction in a very small region
  • 00:35:19
    of the flow we have compressible
  • 00:35:21
    turbulence we have flow which is
  • 00:35:23
    creeping like in the viscous part the
  • 00:35:25
    turbulent boundary where near the wall
  • 00:35:27
    where the flow is essentially zero the
  • 00:35:29
    velocities and very
  • 00:35:30
    very high-speed flow not far away from
  • 00:35:32
    the law where the Mach number might be
  • 00:35:33
    10 or 20 in this case you might have
  • 00:35:36
    flow at the wall which is about Mach
  • 00:35:37
    number zero and away from the wall might
  • 00:35:39
    be about Mach number 20 and so we can
  • 00:35:42
    return to our generalized form of the
  • 00:35:44
    conservation Knab your Stokes equations
  • 00:35:46
    which we shown it's a vector form in our
  • 00:35:49
    fluid dynamics part of the class well a
  • 00:35:51
    partial u partial T plus partial F
  • 00:35:53
    partial X post partial G partial x
  • 00:35:55
    equals zero so here you'll see something
  • 00:35:58
    very simple we can see of course a
  • 00:36:00
    vector form the equations which we'll
  • 00:36:02
    try and operate on now will expand in an
  • 00:36:05
    explicit form for U of n plus 1 which
  • 00:36:08
    would be the solution of the time sub n
  • 00:36:10
    plus 1 a form of the Taylor series upon
  • 00:36:12
    this particular first set of equations
  • 00:36:15
    by expanding the first term in the
  • 00:36:17
    Taylor series you'll find what we I've
  • 00:36:19
    written on the bottom of the page a 27
  • 00:36:21
    well of U of n plus 1 will go as U of n
  • 00:36:24
    plus delta-t
  • 00:36:25
    of partial U of n plus s a partial T
  • 00:36:28
    plus delta T squared over 2 parcel to U
  • 00:36:31
    of M plus SB partial T over squared over
  • 00:36:34
    2 plus higher order terms
  • 00:36:35
    so you see I've done something a little
  • 00:36:37
    bit strange here I've kept the second
  • 00:36:39
    order term the first order term the
  • 00:36:41
    zeroth order term and I've thrown away
  • 00:36:42
    the however terms I truncated them and
  • 00:36:45
    I've done the expansion about N and U of
  • 00:36:47
    M plus 1 with delta T now the new time
  • 00:36:51
    levels on the right hand side that's my
  • 00:36:53
    choice to write them that way I've
  • 00:36:55
    essentially written with something like
  • 00:36:57
    an implicit method is unwrapping in time
  • 00:36:59
    level n plus si and + SB so instead of
  • 00:37:03
    an it it's a well defined sort of like
  • 00:37:06
    midpoint method I'm actually choosing
  • 00:37:08
    two particular midpoints between n time
  • 00:37:12
    level N and n plus 1 and I call these SA
  • 00:37:14
    and SB as a operates the odd time
  • 00:37:19
    derivative and SB operates on the even
  • 00:37:21
    time derivative let's see why we've made
  • 00:37:23
    that choice here now we can split and
  • 00:37:27
    rewrite this first and second order time
  • 00:37:31
    derivative terms the right-hand side the
  • 00:37:33
    first one I've written as partial U of n
  • 00:37:35
    plus si over partial T goes as partial U
  • 00:37:38
    n partial T plus si of partial Delta u n
  • 00:37:41
    plus one
  • 00:37:42
    over partial T and the ranges of si must
  • 00:37:46
    be bounded between 0 & 1 I can rewrite
  • 00:37:49
    the second-order equation here in the
  • 00:37:52
    same methodology also the coefficient s
  • 00:37:55
    B is between 0 & 1 so I redefine these
  • 00:37:58
    and I can take these and insert and
  • 00:37:59
    recover my of course Taylor expansion
  • 00:38:01
    here I've written Delta U of n plus 1
  • 00:38:04
    will go as U of n plus 1 minus u n so
  • 00:38:06
    it's just a shorthand notation between
  • 00:38:08
    the two night-time level solutions
  • 00:38:10
    remember these are vectors for row
  • 00:38:12
    momentum and energy we can now
  • 00:38:14
    substitute these equations in the
  • 00:38:16
    previous one and we will find U of n
  • 00:38:19
    plus 1 goes as unit and plus delta T
  • 00:38:21
    plus the SI term plus delta T squared
  • 00:38:23
    over 2 plus the SB term that's what this
  • 00:38:25
    middle equation is we want to do this
  • 00:38:28
    through jacobians and of course we want
  • 00:38:31
    to solve
  • 00:38:31
    perhaps in a finite difference approach
  • 00:38:33
    to a computational domain from the
  • 00:38:36
    physical space which has uniform spacing
  • 00:38:38
    course so we'll introduce the jacobians
  • 00:38:39
    once we do that we can introduce them in
  • 00:38:42
    terms of the convection diffusion and
  • 00:38:44
    diffusion gradient terms and write the
  • 00:38:46
    first and second derivatives of the
  • 00:38:47
    cancer variables in this form this is
  • 00:38:50
    simply our original equation with the
  • 00:38:51
    right hand side moved the left hand side
  • 00:38:53
    moved to the right and then we'll also
  • 00:38:56
    find a second-order term with our
  • 00:38:58
    definitions above you can combine these
  • 00:39:00
    two particular derivatives and find this
  • 00:39:02
    intermediate equation we'll find partial
  • 00:39:04
    T partial T squared goes as partial
  • 00:39:07
    partial X I of AI plus bi times partial
  • 00:39:10
    FJ partial XJ plus partial GJ partial XJ
  • 00:39:13
    plus this mixed derivative we can then
  • 00:39:17
    substitute these terms into our form of
  • 00:39:20
    U plus M of 1 equals u n plus our higher
  • 00:39:24
    order terms and we'll find this large
  • 00:39:27
    equation it will be for Delta of U and
  • 00:39:31
    plus 1 will go as delta T times all
  • 00:39:33
    these terms which you previously
  • 00:39:35
    substituted but you'll see in this
  • 00:39:37
    equation we have the typical flux terms
  • 00:39:39
    on the right hand side plus an sa times
  • 00:39:41
    the partial derivative of the flux terms
  • 00:39:43
    at n plus 1 plus a second delta T
  • 00:39:47
    squared over 2 of the first order
  • 00:39:50
    derivatives plus the second order time
  • 00:39:54
    derivative terms
  • 00:39:55
    coefficient SB with the flux terms plus
  • 00:39:58
    higher order terms so these
  • 00:40:01
    substitutions and changes the variations
  • 00:40:06
    with Espeon SP are simply resubstitute
  • 00:40:08
    in the previous equation and software
  • 00:40:10
    for the time LaLanne plus one that's a
  • 00:40:12
    lot of work and we've done this in a
  • 00:40:14
    very careful way to find an implicit
  • 00:40:18
    scheme of course for f DB why did we do
  • 00:40:22
    this and I'll try to explain this
  • 00:40:24
    physically now the parameters SA and SB
  • 00:40:28
    will have certain physical analogies and
  • 00:40:31
    we'll find them and set them based on
  • 00:40:34
    the current flow solution so SB and si
  • 00:40:37
    are not constant coefficients they will
  • 00:40:39
    vary depending on the local flow
  • 00:40:41
    properties this is a kind of model if
  • 00:40:44
    you will now let's talk about si first
  • 00:40:46
    which is of course operating on the
  • 00:40:49
    first time derivative if si si shows you
  • 00:40:52
    two temporal changes of convection it
  • 00:40:54
    may be viewed from changes in Mach
  • 00:40:56
    number between adjacent nodal points we
  • 00:40:59
    will set si we'd apply that there's no
  • 00:41:01
    changes in the convection fluctuations
  • 00:41:03
    let's look at an example for the
  • 00:41:05
    variation of si for a shock tube problem
  • 00:41:08
    a shock tube problem of course is where
  • 00:41:10
    we have a long pipe and we have a
  • 00:41:11
    high-pressure gas separated by a
  • 00:41:13
    low-pressure gas a diaphragm separates
  • 00:41:16
    them and an explosion occurs which blows
  • 00:41:18
    apart the metal diaphragm and the high
  • 00:41:20
    pressure region is next to low pressure
  • 00:41:22
    region this creates a shock wave which
  • 00:41:24
    travels to the right in the system from
  • 00:41:26
    high pressure to low pressure
  • 00:41:27
    of course the shock will eventually
  • 00:41:30
    reach the end of the tube and will
  • 00:41:31
    reflect here we show particular
  • 00:41:33
    solutions of the shock tube problem we
  • 00:41:37
    also show relative velocities pressures
  • 00:41:40
    and temperatures you can see in this
  • 00:41:43
    particular case si will vary depending
  • 00:41:47
    on the particular solution of where it
  • 00:41:49
    is in the system
  • 00:41:50
    let's look instead of a shock on a bound
  • 00:41:52
    Euler problem for a boundary layer flow
  • 00:41:54
    si would be one that is dependent on the
  • 00:41:57
    fluctuations of diffusion such as the
  • 00:41:59
    boundary layer flows therefore si would
  • 00:42:01
    be dependent on something like the
  • 00:42:03
    changes of the reynolds number of the
  • 00:42:04
    Peck
  • 00:42:05
    a great working model for sa could be
  • 00:42:07
    formed and so fdv might actually have
  • 00:42:10
    models that vary sa depending on of
  • 00:42:14
    course the Flex vectors F and G it would
  • 00:42:17
    be very beneficial to find general
  • 00:42:18
    models for SA and SB we've only showed
  • 00:42:21
    you two particular models si for a shock
  • 00:42:23
    to problem and SB for of course a
  • 00:42:26
    boundary layer problem but this is
  • 00:42:28
    rather cumbersome in that we have to
  • 00:42:29
    create entire models just to vary SA and
  • 00:42:32
    SB through the flow field we'll try and
  • 00:42:35
    seek out just general models and how
  • 00:42:37
    might we do that well to provide these
  • 00:42:40
    variation of these parameters and
  • 00:42:42
    changes with convection diffusion we
  • 00:42:44
    would need to know something about the
  • 00:42:46
    current flow field so we would need to
  • 00:42:48
    make our models dependent on current
  • 00:42:50
    flow field which is kind of like this
  • 00:42:51
    equation here at the top of the page
  • 00:42:53
    we note that SA and SB to find accurate
  • 00:42:56
    solutions for different kinds of flow
  • 00:42:58
    fields have certain physical properties
  • 00:42:59
    you might know that si with Delta F will
  • 00:43:03
    be called s 1 si with Delta G will be s
  • 00:43:06
    3 SB with Delta G will be s 2 and s B
  • 00:43:09
    with Delta G will be s 4 so now we
  • 00:43:12
    divided SA and SB into 4 different
  • 00:43:15
    parameters depending on if it's
  • 00:43:17
    operating on the fluxes f or g so s 1
  • 00:43:20
    will look at his first-order convection
  • 00:43:22
    fdv parameter s 3 will be the
  • 00:43:24
    first-order diffusion ftv parameter and
  • 00:43:26
    2 & 4 will be the second-order terms for
  • 00:43:29
    the convection diffusion parameters
  • 00:43:30
    let's illustrate these particular
  • 00:43:33
    breakdown of s1 through s4 for these
  • 00:43:35
    parameters the convection parameters
  • 00:43:38
    will call s 1 and s 2 and parameters
  • 00:43:40
    that are altered by diffusion physics
  • 00:43:42
    and mechanisms will call s 3 S 4 s 1 and
  • 00:43:45
    s 2 are going to be more so dependent on
  • 00:43:47
    Mach number and s 3 and s 4 are going to
  • 00:43:49
    be more so dependent on Reynolds number
  • 00:43:50
    and so will look at the first order
  • 00:43:52
    parameters first we can define simple
  • 00:43:56
    models based on particular Mach numbers
  • 00:43:58
    for s 1 and s 2 here's one functional
  • 00:44:01
    form for s 1 and 1 functional form for s
  • 00:44:03
    to the first order F TV parameters can
  • 00:44:07
    be shown on the right for s 1 and s 2
  • 00:44:09
    with Y axis for F TV parameters s 3 and
  • 00:44:12
    s for diffusion based so for diffusion
  • 00:44:15
    based parameters these simple models are
  • 00:44:18
    chosen
  • 00:44:18
    based on say minimum zero or one
  • 00:44:20
    depending on the Reynolds numbers so you
  • 00:44:22
    can see it s1 has two are depending on
  • 00:44:25
    Mach numbers in a very very similar form
  • 00:44:27
    is given for say s3 s4 which are
  • 00:44:30
    dependent on rounds number and the
  • 00:44:32
    peclet number let's look at one
  • 00:44:34
    particular variation for a shock in a
  • 00:44:37
    corner here the flow moves from left to
  • 00:44:40
    right an initial in LOC Inlet shock
  • 00:44:43
    forms and then in the corner another
  • 00:44:45
    oblique shock form so we have two
  • 00:44:47
    oblique shocks which are being turned of
  • 00:44:49
    course by the flow so the flows pretty
  • 00:44:52
    much uniform in before the shock after
  • 00:44:54
    the shock and after the corner shock and
  • 00:44:57
    the corner shocks and the inland shock
  • 00:44:59
    have extremely large gradients so we
  • 00:45:02
    would expect s1 and s2 to be zero in the
  • 00:45:05
    regions between the shocks at the shocks
  • 00:45:09
    themselves who would probably have very
  • 00:45:10
    very high values that's one most two for
  • 00:45:13
    the diffusion parameters we would expect
  • 00:45:15
    s3 and s4 to be zero and we might have a
  • 00:45:19
    rotational flow near the shock where we
  • 00:45:21
    would of course have s3 and s4 as one so
  • 00:45:24
    by looking at this simple problem we can
  • 00:45:26
    estimate what the values are and indeed
  • 00:45:28
    s1 s2 s3 and s4 are found as a matter of
  • 00:45:31
    the solution as a function of space for
  • 00:45:34
    steady solutions filed through F DV what
  • 00:45:38
    s1 through s4 essentially do are
  • 00:45:40
    parameters which scale and weight
  • 00:45:42
    certain terms in the equations of motion
  • 00:45:44
    if say the diffusion terms are dominant
  • 00:45:47
    or the convection terms are dominant and
  • 00:45:51
    those go of course as the first and
  • 00:45:53
    second order derivatives in the Taylor
  • 00:45:56
    expansion of U respectively let's
  • 00:45:58
    summarize our full form of the f DB
  • 00:46:01
    equations and so if you have a research
  • 00:46:02
    solver or commercial solver and you
  • 00:46:05
    select f DB for these very high speed
  • 00:46:07
    flows then these an equation you'll be
  • 00:46:09
    solving the change in the vector U which
  • 00:46:13
    is the field variables at time plus one
  • 00:46:16
    plus delta T times s 1 plus s3 minus
  • 00:46:19
    delta T squared of the s 2 s 4
  • 00:46:21
    parameters plus delta T of the original
  • 00:46:25
    fluxes minus delta T squared over two of
  • 00:46:28
    the modified flexions plus hard our
  • 00:46:30
    terms will go to zero
  • 00:46:32
    so we have a single vector equation for
  • 00:46:34
    the continuity momentum energy equations
  • 00:46:37
    with weighting factors s1 through s4
  • 00:46:39
    which are dependent on the flow field
  • 00:46:41
    solution you can always ask them main s1
  • 00:46:44
    through s4 in advance as part of the
  • 00:46:45
    initial condition but it's honestly best
  • 00:46:47
    to let them be calculated by the initial
  • 00:46:50
    condition itself of densities velocities
  • 00:46:52
    pressures and temperatures so you can
  • 00:46:53
    see the physical phenomena is in these
  • 00:46:55
    flows are actually dictated by the F DV
  • 00:46:57
    parameters themselves and our inherent
  • 00:46:58
    models now what we haven't done is
  • 00:47:02
    discretize this set of equations
  • 00:47:04
    necessarily with say a finite difference
  • 00:47:06
    method of finite element method or
  • 00:47:08
    finite volume method so we would have to
  • 00:47:10
    take this vector equation and discretize
  • 00:47:12
    it with one of these three methods once
  • 00:47:15
    we do that we would also have to apply
  • 00:47:17
    to balance model if we really care about
  • 00:47:20
    modeling turbulence and high-speed flows
  • 00:47:22
    which we'll discuss later in the class
  • 00:47:23
    so this is in the end of the story of F
  • 00:47:26
    DV it's only the beginning and you can
  • 00:47:28
    imagine how a researcher would have to
  • 00:47:30
    take this equation discretize it perhaps
  • 00:47:33
    of a finite element method for high
  • 00:47:35
    order discretization and then apply it
  • 00:47:38
    and even modify these equations further
  • 00:47:40
    to have additional equations in the
  • 00:47:43
    vector form with a turbulence model this
  • 00:47:45
    could actually take years of someone's
  • 00:47:47
    life and even additional years to
  • 00:47:50
    program it in a contemporary CFD solver
  • 00:47:52
    so it's impossible to learn and
  • 00:47:54
    understand any of these massive methods
  • 00:47:56
    especially flow field dependent
  • 00:47:58
    variation in a single class you're
  • 00:48:00
    probably feeling completely overwhelmed
  • 00:48:01
    and that's okay
  • 00:48:03
    it's just a tip of the iceberg of the
  • 00:48:05
    field of CFD which of course is perhaps
  • 00:48:07
    seen as a mild why entity methodology
  • 00:48:11
    and a field let's take away some
  • 00:48:13
    physical points of the fdv method the
  • 00:48:15
    first-order ftv parameters s1 and s3
  • 00:48:18
    will control all the high gradient flow
  • 00:48:20
    phenomena such as shocks and turbulence
  • 00:48:21
    and so where we have high gradients of
  • 00:48:23
    shock waves or very intense turbulence
  • 00:48:25
    s1 and s3 Multani these parameters will
  • 00:48:28
    be calculated through changes of local
  • 00:48:30
    Mach numbers and Reynolds numbers within
  • 00:48:32
    each element of the flow field there
  • 00:48:35
    could be viewed as actual local elements
  • 00:48:38
    of the flow the contours of s1 and s3
  • 00:48:40
    will resemble the axial flow field
  • 00:48:42
    features themselves
  • 00:48:44
    frig's
  • 00:48:45
    and the shock problem contour of s1 and
  • 00:48:48
    s3 would very much like go along these
  • 00:48:51
    types of rotational flows the fact in
  • 00:48:54
    that contours of s1 and 3 s 3 will
  • 00:48:56
    resemble say Mach number density
  • 00:48:58
    contours has been done demonstrating
  • 00:49:00
    this previous example the role in s1 s3
  • 00:49:03
    of course is to provide computational
  • 00:49:05
    accuracy the second order STV parameters
  • 00:49:08
    are a little bit different they will
  • 00:49:09
    provide exponentially proportional terms
  • 00:49:13
    to the first order SVD parameters
  • 00:49:15
    however their role is more to adequately
  • 00:49:18
    provide computational stability and
  • 00:49:20
    artificial viscosity they were
  • 00:49:22
    introduced as you recall through the
  • 00:49:23
    Taylor series of expansion on a second
  • 00:49:25
    order terms if s1 is 0 these terms would
  • 00:49:29
    represent convection terms this applies
  • 00:49:32
    that if s1 is approximately 0 lumen the
  • 00:49:34
    convection is small this might be a
  • 00:49:37
    rather stationary type flow with a low
  • 00:49:39
    velocity the computational scheme will
  • 00:49:41
    automatically be altered to take this
  • 00:49:43
    effect in account when the governing
  • 00:49:45
    equations are predominantly parabolic
  • 00:49:47
    that is very low speed or transonic if
  • 00:49:51
    s3 is approximately zero in our
  • 00:49:53
    solutions it'll likely be a hyperbolic
  • 00:49:55
    type system of equations and the flow
  • 00:49:57
    would be very very high-speed the scheme
  • 00:50:00
    will automatically switch to something
  • 00:50:01
    like an Euler equations where the
  • 00:50:03
    equations are generally hyperbolic and
  • 00:50:05
    are a very high speed hypersonic flow
  • 00:50:07
    without many terms of turbulence or
  • 00:50:10
    shock 6 era if we look at our solution
  • 00:50:12
    and s1 and s3 are not 0 and they're
  • 00:50:16
    rather dominant then we'll see that
  • 00:50:19
    these indicates that the simulation is
  • 00:50:21
    rather a mixed hyperbolic parabolic and
  • 00:50:23
    elliptic in nature which of course is
  • 00:50:26
    like an ave Stokes system of equations
  • 00:50:28
    and the convection diffusion terms will
  • 00:50:30
    be balanced so if s1 s2 and s3 s4 are
  • 00:50:36
    not 0 and they're equal will of course
  • 00:50:38
    have a rather good balance between
  • 00:50:40
    convection and diffusion terms these all
  • 00:50:44
    these cases can actually be seen
  • 00:50:46
    everywhere in a type of flow and this
  • 00:50:48
    will always be the case for
  • 00:50:50
    incompressible flows at low speeds you
  • 00:50:52
    would not typically choose F DB for a
  • 00:50:54
    purely incompressible flow that would be
  • 00:50:56
    a waste of computational resources and
  • 00:50:58
    you should look at something like the P
  • 00:50:59
    scheme or simple or some simpler scheme
  • 00:51:02
    which we looked at in a previous class
  • 00:51:04
    unique properties of the FTC scheme will
  • 00:51:08
    be capable to control the presser
  • 00:51:10
    oscillations adequately without
  • 00:51:11
    resorting to separately hyperbolic terms
  • 00:51:13
    like the Poisson equation to pressure
  • 00:51:15
    Corrections
  • 00:51:16
    that's one wonderful thing about the F
  • 00:51:18
    DV scheme is we won't see pressure
  • 00:51:19
    oscillations like we had in the other
  • 00:51:22
    schemes and you don't have to resort to
  • 00:51:24
    some sort of like intermediate grid
  • 00:51:26
    overlaps or for example like I plus 1/2
  • 00:51:29
    or I minus 1/2 as we saw in for example
  • 00:51:33
    the simple scheme now the F DV scheme
  • 00:51:35
    could be applied to incompressible flows
  • 00:51:37
    if we delicately balance s1 and s3 and
  • 00:51:41
    of course it happens automatically and
  • 00:51:43
    say the near wall region of that
  • 00:51:45
    turbulent boundary there where this flow
  • 00:51:47
    is high-speed the flow is completely
  • 00:51:49
    incompressible that is the global Mach
  • 00:51:50
    number is zero but there still flow
  • 00:51:52
    fluctuations then of course s1 will be 1
  • 00:51:54
    then will set s1 equal to one explicitly
  • 00:51:57
    and we'll let the variation the Primmer
  • 00:51:58
    s3 be determined through the flow solver
  • 00:52:01
    at the end of the day all these floats
  • 00:52:03
    terms and interactions between
  • 00:52:05
    convection diffusion will happen
  • 00:52:06
    automatically if we use a general model
  • 00:52:08
    for the for fdv parameters in this class
  • 00:52:12
    we summarized some major algorithms for
  • 00:52:15
    numerical solutions of incompressible
  • 00:52:16
    and compressible flows these are
  • 00:52:18
    obviously a little bit beyond an
  • 00:52:20
    introduction to CFD class and are often
  • 00:52:22
    taught more so in depth in Graduate CFD
  • 00:52:24
    classes nonetheless we have looked at
  • 00:52:27
    some workhorse algorithms so to speak
  • 00:52:30
    for many advanced commercial and
  • 00:52:32
    industrial solvers you will see a CM
  • 00:52:35
    simple piezo and variations like
  • 00:52:38
    compressible piezo and maybe
  • 00:52:40
    FDB in many of these types of solvers
  • 00:52:43
    incompressible flows are certainly
  • 00:52:45
    handled excellently by the ACM simple
  • 00:52:48
    and piezo algorithm if you have a
  • 00:52:50
    compressibility in your flow your solver
  • 00:52:52
    should handle of course these
  • 00:52:54
    fluctuations for example through the
  • 00:52:56
    modification of piezo preconditioning
  • 00:52:59
    techniques for creeping flows which are
  • 00:53:00
    incompressible or compressible and of
  • 00:53:03
    course the very famous FDB technique for
  • 00:53:06
    very high speed flows and hypersonics
  • 00:53:08
    and high speed SuperSonics now there's
  • 00:53:10
    many other algorithms
  • 00:53:12
    in fact we could have a whole class just
  • 00:53:13
    talking about all the different types of
  • 00:53:15
    CFD algorithms I was careful and just
  • 00:53:17
    choose three or four of the most popular
  • 00:53:20
    ones which you'll see as choices in the
  • 00:53:21
    solvers from this class try and take
  • 00:53:25
    away the overall approach and realize
  • 00:53:27
    what's happening the solvers and why
  • 00:53:29
    some of these choices are made for
  • 00:53:30
    example to eliminate the so called
  • 00:53:32
    checkerboard problem pressure and
  • 00:53:34
    compressible flows or to handle
  • 00:53:36
    seamlessly the different types of
  • 00:53:38
    dominant terms are convection and
  • 00:53:40
    diffusion in a high speed turbulent
  • 00:53:42
    boundary layer next time don't try and
  • 00:53:44
    ground ourselves in stability analysis
  • 00:53:47
    for simpler CFD schemes the same
  • 00:53:49
    stability analysis can be applied to
  • 00:53:51
    these complicated schemes but it's
  • 00:53:53
    usually not performed and only examine
  • 00:53:55
    numerically then we'll talked about
  • 00:53:56
    residual and convergence and look at the
  • 00:53:58
    so-called l1 and l2 norms if we run
  • 00:54:00
    these schemes or schemes we previously
  • 00:54:02
    talked about then we will look at how
  • 00:54:04
    convergence is achieved and how we
  • 00:54:06
    measure air well then formally define
  • 00:54:08
    and look at convergence and give
  • 00:54:10
    examples thank you very much for your
  • 00:54:12
    time today I'm professor Steve Miller
タグ
  • CFD
  • computational fluid dynamics
  • solver algorithms
  • incompressible flow
  • compressible flow
  • artificial compressibility
  • SIMPLE
  • PISO
  • preconditioning
  • FDB