[CFD] Conservative, Advective & Material Derivative forms of the Navier-Stokes Equations

00:32:20
https://www.youtube.com/watch?v=ljdv4T2U464

概要

TLDRThis talk provides an overview of the various forms of the Navier-Stokes equations and other transport equations, clarifying their differences and applications. Using a swimming pool temperature gradient as an example, the speaker explains concepts like the material derivative and the relationship between Lagrangian and Eulerian descriptions. The talk emphasizes the equivalence of different forms, including advective and conservative forms, and their significance in computational fluid dynamics (CFD). The speaker encourages viewers to understand these differences for accurate application in their work.

収穫

  • 📚 Understanding different forms of Navier-Stokes equations is crucial for accurate application.
  • 🌊 The swimming pool example illustrates temperature gradients and sensor movement.
  • 🔄 The material derivative captures both spatial and temporal changes.
  • ⚖️ Lagrangian and Eulerian forms serve different analytical purposes.
  • 🔍 The conservative form is preferred in CFD for its ease of application.
  • 📏 The continuity equation plays a key role in deriving relationships between forms.
  • 🧮 All forms of the equations are equivalent; the choice depends on context.
  • 💡 Knowing these forms enhances confidence in citing equations in research.
  • 📝 The talk encourages further questions and clarifications on CFD topics.

タイムライン

  • 00:00:00 - 00:05:00

    The talk introduces various forms of the Navier-Stokes equations and transport equations, emphasizing the importance of understanding these differences for accurate writing in academic papers. The speaker aims to clarify these forms to prevent typographical errors in research work.

  • 00:05:00 - 00:10:00

    A simple example of a swimming pool with a temperature gradient is used to illustrate the concept of temperature measurement. The speaker explains how moving a temperature sensor through the pool affects the readings, highlighting the relationship between sensor movement and temperature change over time.

  • 00:10:00 - 00:15:00

    The discussion extends to three-dimensional movement of the sensor, leading to a formula that relates the rate of temperature change to the velocity of the sensor and the spatial temperature gradient. The speaker emphasizes the importance of the correct order of operations in vector notation for accurate calculations.

  • 00:15:00 - 00:20:00

    The speaker introduces the concept of the material derivative, which accounts for both spatial and temporal variations in temperature. This derivative is crucial for deriving the Navier-Stokes equations, allowing for a more straightforward approach compared to traditional methods.

  • 00:20:00 - 00:25:00

    The derivation of the Navier-Stokes equations is presented using the material derivative, focusing on a fluid parcel's momentum and the effects of external forces. The speaker explains how this approach simplifies the derivation process while maintaining the integrity of the equations.

  • 00:25:00 - 00:32:20

    The talk concludes by summarizing the equivalence of different forms of the Navier-Stokes equations, including advective and conservative forms. The speaker emphasizes the importance of understanding these forms for practical applications in computational fluid dynamics (CFD) and encourages viewers to engage with the content for further clarification.

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ビデオQ&A

  • What are the different forms of the Navier-Stokes equations?

    The different forms include Lagrangian derivative form, conservative form, and advection form.

  • What is the material derivative?

    The material derivative represents the rate of change of a quantity as measured by a moving sensor, accounting for both spatial and temporal variations.

  • How do the Lagrangian and Eulerian forms differ?

    Lagrangian form tracks fluid parcels, while Eulerian form analyzes fixed volumes as fluid flows through them.

  • Why is the conservative form preferred in CFD?

    The conservative form is easier to apply in finite volume methods due to its compatibility with the Divergence Theorem.

  • What is the significance of the temperature gradient in the example?

    The temperature gradient illustrates how the measured temperature changes as a sensor moves through different temperature regions.

  • Can the Navier-Stokes equations be derived using the material derivative?

    Yes, the material derivative allows for a quicker derivation of the Navier-Stokes equations.

  • What is the relationship between advective and conservative forms?

    Both forms are equivalent; the choice depends on the context and method of solution.

  • What is the role of the continuity equation in these derivations?

    The continuity equation helps simplify terms when deriving the relationship between different forms of the Navier-Stokes equations.

  • How can I apply these concepts in my own work?

    Understanding these forms will help you accurately cite and utilize the Navier-Stokes equations in your research.

  • What should I do if I have more questions about CFD equations?

    You can leave comments or questions for further clarification on specific topics.

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  • 00:00:01
    if you're writing your thesis a
  • 00:00:04
    conference paper or a journal paper or
  • 00:00:06
    even just looking through the cfd user
  • 00:00:08
    manuals you may have come across
  • 00:00:10
    different forms of the navier Stokes
  • 00:00:12
    equations and the other transport
  • 00:00:15
    equations some of the Common forms which
  • 00:00:17
    you might have seen include the
  • 00:00:19
    lagrangian derivative form a
  • 00:00:21
    conservative form and an advection form
  • 00:00:24
    of the same transport equations and it's
  • 00:00:27
    often not clear what the differences are
  • 00:00:29
    between these different forms of the
  • 00:00:31
    same equation
  • 00:00:33
    what I'm going to be doing for you in
  • 00:00:35
    this talk is going through the different
  • 00:00:37
    forms of the transport equations so that
  • 00:00:40
    you understand the differences between
  • 00:00:41
    them and you can make sure you don't put
  • 00:00:43
    any typos in your paper when you're
  • 00:00:45
    writing it out
  • 00:00:46
    so if you're going to be writing these
  • 00:00:48
    equations down this talk is going to be
  • 00:00:50
    really useful for you I'm going to go
  • 00:00:52
    through all of the different forms sit
  • 00:00:54
    back and let's get into the talk
  • 00:00:58
    the easiest way to understand the
  • 00:01:00
    different forms of the navi Stokes
  • 00:01:02
    equations is to start with a simple
  • 00:01:05
    example and the example I'm going to use
  • 00:01:07
    is imagine a swimming pool or a large
  • 00:01:11
    volume of water and this volume of water
  • 00:01:15
    is contained in a container and for some
  • 00:01:18
    reason which we don't need to think
  • 00:01:20
    about
  • 00:01:21
    the pool of water is cold at one end and
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    hot at the other end so there's a
  • 00:01:27
    gradient of temperature from one end to
  • 00:01:29
    the other end of the pool I'm not
  • 00:01:31
    considering any boundary layers heated
  • 00:01:33
    surfaces or variation in the vertical
  • 00:01:35
    Direction just a simple one-dimensional
  • 00:01:38
    case where we have a cold end at one end
  • 00:01:41
    and a hot end at the other end this is
  • 00:01:44
    the example we're going to use for
  • 00:01:45
    understanding the different forms of the
  • 00:01:47
    navier Stokes equations
  • 00:01:50
    and what I want you to think about is
  • 00:01:52
    placing a temperature sensor of some
  • 00:01:55
    kind like a thermometer for example in
  • 00:01:58
    the pool at a given location
  • 00:02:00
    and in this example the temperature of
  • 00:02:03
    the pool itself is not varying with time
  • 00:02:06
    so the thermometer will read a constant
  • 00:02:09
    temperature Through Time Pool has a
  • 00:02:12
    constant temperature of course in
  • 00:02:14
    reality there would be some small
  • 00:02:16
    variations in temperature around the
  • 00:02:18
    measured value but I'm not going to be
  • 00:02:20
    considering those today all I want you
  • 00:02:22
    to think about is if we place the sensor
  • 00:02:24
    somewhere in the pool the temperature
  • 00:02:26
    will be constant with time
  • 00:02:29
    now what happens if we move the sensor
  • 00:02:33
    this is the key idea that I want you to
  • 00:02:35
    think about if we take that sensor which
  • 00:02:38
    is initially at the cold end of the pool
  • 00:02:39
    and then move the sensor through the
  • 00:02:42
    water to the hot end of the pool I want
  • 00:02:44
    you to think about physically moving
  • 00:02:46
    that sensor yourself and if we look on
  • 00:02:49
    the screen where the data is shown for
  • 00:02:51
    what the temperature measures we will of
  • 00:02:53
    course see that the measured temperature
  • 00:02:55
    increases with time because we're moving
  • 00:02:58
    that sensor from the cold end of the
  • 00:03:00
    pool all the way through to the hot end
  • 00:03:02
    of the pool
  • 00:03:04
    so even though the temperature of the
  • 00:03:06
    pool itself is staying constant in time
  • 00:03:08
    because we're moving the sensor through
  • 00:03:11
    the pool the measured temperature that
  • 00:03:13
    the sensor sees does change with time
  • 00:03:17
    and following that same idea if we move
  • 00:03:20
    the sensor faster so we really move it
  • 00:03:23
    quickly through the pool the temperature
  • 00:03:24
    measured by the sensor will change more
  • 00:03:27
    rapidly and you can see that there in
  • 00:03:29
    the Curve and I really want you to think
  • 00:03:31
    about you actually doing this yourself
  • 00:03:33
    taking the sensor moving it really fast
  • 00:03:36
    through the pool and the temperature
  • 00:03:38
    Trace that you'll see on the screen of
  • 00:03:40
    the sensor is that the temperature
  • 00:03:41
    varies more rapidly with time
  • 00:03:45
    and we can do a similar thought exercise
  • 00:03:47
    where if we had two pulls one which is
  • 00:03:51
    cold at one end and hot at the other end
  • 00:03:52
    and then we had a second pool which was
  • 00:03:55
    a lot colder at one end and a lot hotter
  • 00:03:57
    at the other end it has a greater
  • 00:04:00
    spatial temperature gradient than if we
  • 00:04:02
    move the sensor at the same speed
  • 00:04:04
    through both of the pools of course the
  • 00:04:07
    pool with the steeper temperature
  • 00:04:08
    gradient is going to see a more rapid
  • 00:04:11
    change in temperature in time as we're
  • 00:04:14
    moving through a steeper temperature
  • 00:04:16
    gradient and these are the key ideas
  • 00:04:18
    that I want you to think about
  • 00:04:21
    we can actually bring these ideas
  • 00:04:23
    together with a very simple equation
  • 00:04:25
    we want to be thinking about the
  • 00:04:27
    measured rate of change of temperature
  • 00:04:29
    so what is the gradient of temperature
  • 00:04:31
    that's seen on the screen of the sensor
  • 00:04:34
    as we move it through the pool well it
  • 00:04:37
    turns out that if you really think about
  • 00:04:39
    moving along the spatial temperature
  • 00:04:42
    gradient in the pool at some velocity U
  • 00:04:44
    it turns out that the rate of measured
  • 00:04:48
    temperature change by the sensor is just
  • 00:04:50
    going to be equal to the spatial
  • 00:04:52
    temperature gradient dtdx multiplied by
  • 00:04:56
    the speed U that we move through the
  • 00:04:58
    pool and of course you can verify or
  • 00:05:01
    check this by just looking at the units
  • 00:05:03
    DT DT is a units of kelvins per second
  • 00:05:07
    or degrees per second and then that's
  • 00:05:09
    going to be equal to Velocity meters per
  • 00:05:11
    second multiplied by the spatial
  • 00:05:13
    temperature gradient Kelvin per meter so
  • 00:05:16
    that's our simple formula the measured
  • 00:05:18
    rate of change of temperature with time
  • 00:05:20
    is going to be equal to the speed
  • 00:05:21
    multiplied by the spatial temperature
  • 00:05:23
    gradient
  • 00:05:25
    now that's in one dimension but what if
  • 00:05:28
    we took our sensor and moved it in some
  • 00:05:31
    3D Direction in the pool if we're not
  • 00:05:33
    moving from left to right maybe we move
  • 00:05:35
    up to down or we move diagonally can
  • 00:05:38
    actually extend that previous equation
  • 00:05:40
    that we had there dtdt is again going to
  • 00:05:43
    be equal to the X component of the
  • 00:05:45
    Velocity Vector U multiplied by DT DX
  • 00:05:48
    and then we have the same contributions
  • 00:05:51
    in the Y and Z directions as well but
  • 00:05:55
    it's the same formula because of course
  • 00:05:57
    you can verify this if you were to move
  • 00:06:00
    parallel or move into the screen so
  • 00:06:02
    we're not moving in the X Direction but
  • 00:06:04
    we're moving with some velocity
  • 00:06:05
    component V then the measured rate of
  • 00:06:08
    change of temperature would be equal to
  • 00:06:09
    the speed that we move in that direction
  • 00:06:12
    multiplied by the temperature gradient
  • 00:06:15
    in that direction as well
  • 00:06:17
    and we can simplify this equation into
  • 00:06:20
    Vector form by noticing of course that U
  • 00:06:23
    V and W are the components of the
  • 00:06:25
    Velocity gradient and we can rewrite
  • 00:06:26
    this as the dot product of the Velocity
  • 00:06:29
    vector and the temperature gradient
  • 00:06:32
    Vector there DT DX dtdy and DT DZ and we
  • 00:06:37
    can also simplify this by rewriting in
  • 00:06:39
    Vector notation using bold U for the
  • 00:06:42
    velocity Vector nabla for the gradient
  • 00:06:45
    vector and then t for the temperature
  • 00:06:47
    field
  • 00:06:48
    so this is our formula equation four the
  • 00:06:52
    measured rate of change of temperature
  • 00:06:54
    on the screen of the sensor is going to
  • 00:06:56
    be equal to the dot product U Dot nabla
  • 00:06:59
    and then multiplied by the temperature
  • 00:07:01
    gradient the temperature field there and
  • 00:07:04
    I wanted to make a quick side point here
  • 00:07:06
    to be very careful with this formula
  • 00:07:08
    that you have the order of the dot
  • 00:07:10
    product the correct way round because of
  • 00:07:12
    course the Divergence of the Velocity
  • 00:07:15
    field nabla.u if you have an
  • 00:07:17
    incompressible flow that's going to be
  • 00:07:19
    equal to zero so this formula is not
  • 00:07:22
    correct you have to make sure to use
  • 00:07:24
    equation four the velocity Vector is
  • 00:07:26
    being we're taking the dot product of
  • 00:07:28
    that with the gradient which is applied
  • 00:07:30
    to the temperature field
  • 00:07:32
    so take care to get the order of the
  • 00:07:34
    operations the right way around there
  • 00:07:38
    now carrying on with this simple example
  • 00:07:40
    we are going to be building towards some
  • 00:07:42
    important formulas soon now let's
  • 00:07:44
    consider the case where we have the
  • 00:07:46
    sensor at a fixed location in the pool
  • 00:07:48
    but the temperature of the pool is
  • 00:07:50
    varying in time so there is some time
  • 00:07:52
    variation in that sensor even if we
  • 00:07:55
    don't move it
  • 00:07:57
    of course if we do both if we have a
  • 00:08:00
    background temperature field that does
  • 00:08:01
    vary in time and we're also moving
  • 00:08:04
    through a spatial temperature gradient
  • 00:08:06
    then we're going to get both of these
  • 00:08:08
    contributions which may look something
  • 00:08:10
    like this diagram here
  • 00:08:12
    and both of these contributions so the
  • 00:08:15
    background variation of the temperature
  • 00:08:17
    in time and the movement through the
  • 00:08:19
    spatial temperature gradient are both
  • 00:08:21
    going to contribute to the measured rate
  • 00:08:24
    of change of temperature so what we see
  • 00:08:27
    on the screen of the sensor is going to
  • 00:08:29
    have both of these contributions one and
  • 00:08:32
    two
  • 00:08:32
    and to make it absolutely clear of the
  • 00:08:35
    difference between the rate of change of
  • 00:08:37
    temperature measured on the screen and
  • 00:08:39
    the rate of change of temperature of the
  • 00:08:42
    background flow itself I've introduced
  • 00:08:44
    this new notation Capital DT by by
  • 00:08:48
    Capital DT this is new notation to make
  • 00:08:51
    it clear that we've got both
  • 00:08:52
    contributions
  • 00:08:54
    and this new notation for the time
  • 00:08:56
    derivative is sometimes called the
  • 00:08:58
    material derivative or the lagrangian
  • 00:09:00
    derivative you can find it described in
  • 00:09:03
    many different ways on the internet but
  • 00:09:05
    the trick and the way to think about
  • 00:09:07
    this new derivative is it's the
  • 00:09:09
    temperature gradient in time measured by
  • 00:09:12
    a moving sensor so we move a sensor
  • 00:09:15
    through a spatial gradient
  • 00:09:17
    that's the temperature that's seen by
  • 00:09:19
    the sensor and this is not the same as
  • 00:09:21
    the change in the background temperature
  • 00:09:23
    in time that's the way to think about
  • 00:09:25
    this and it's called the lagrangian
  • 00:09:28
    derivative because we're moving with the
  • 00:09:30
    object we're not in an inertial or fixed
  • 00:09:33
    reference frame outside of the object
  • 00:09:37
    now that derivation that we've seen
  • 00:09:39
    there is actually very useful for us and
  • 00:09:43
    the reason that it's useful is we can
  • 00:09:45
    use it to derive The navier Stokes
  • 00:09:47
    equations now the most common derivation
  • 00:09:50
    of the navi Stokes equations that you've
  • 00:09:51
    probably seen is the derivation where we
  • 00:09:54
    use a fixed volume and fluid passes
  • 00:09:56
    through it but actually we can do a much
  • 00:09:59
    quicker derivation of the navi Stokes
  • 00:10:01
    equations using this idea of the
  • 00:10:03
    material derivative
  • 00:10:05
    and that's what I'm going to show you in
  • 00:10:07
    the next section we're then going to
  • 00:10:09
    build later to look at the different
  • 00:10:11
    forms of the navierstopes equations and
  • 00:10:13
    ultimately I'm going to show you that
  • 00:10:15
    all of the forms are consistent and we
  • 00:10:17
    have the same equation and you're going
  • 00:10:19
    to understand the differences between
  • 00:10:20
    the two
  • 00:10:22
    so how do we go about this quick and
  • 00:10:24
    easy derivation of the navi Stokes
  • 00:10:26
    equations what I want you to do is to
  • 00:10:28
    consider a parcel of fluid and what I
  • 00:10:31
    mean by a parcel of fluid is a group of
  • 00:10:34
    fluid molecules all together you can
  • 00:10:36
    imagine this as drawing an imaginary box
  • 00:10:38
    around a group of fluid molecules and
  • 00:10:43
    this group of fluid molecules has a mass
  • 00:10:45
    m
  • 00:10:46
    and what I want you to imagine is this
  • 00:10:48
    parcel of fluid moving at some velocity
  • 00:10:51
    U we're going to be following along with
  • 00:10:53
    this parcel of fluid that moves at some
  • 00:10:55
    velocity U
  • 00:10:57
    now if we think of Newton's laws of
  • 00:10:59
    motion of course if there are if there's
  • 00:11:01
    no net external force acting on that
  • 00:11:04
    parcel of fluid molecules it will
  • 00:11:06
    continue to move with the same momentum
  • 00:11:09
    however if there is a net external force
  • 00:11:13
    acting on this parcel of fluid molecules
  • 00:11:15
    that can be from the surrounding fluid
  • 00:11:18
    or it could be an external Force like
  • 00:11:19
    gravity for example then the momentum of
  • 00:11:22
    the fluid parcel will change and the
  • 00:11:25
    rate of change of its momentum will be
  • 00:11:27
    equal to the net external force and of
  • 00:11:29
    course that's Newton's second law and
  • 00:11:32
    the key I want you to think about here
  • 00:11:33
    is because we're moving with the part of
  • 00:11:36
    the parcel we're moving along with it
  • 00:11:37
    we're going to be using the material
  • 00:11:39
    derivative here capital D by capital d t
  • 00:11:45
    and the advantage of using this approach
  • 00:11:47
    in our quick and easy derivation of the
  • 00:11:50
    navier Stokes equations is that the mass
  • 00:11:53
    of our parcel of fluid doesn't change
  • 00:11:56
    now if you think about for example a
  • 00:11:58
    flow field that may be heated there may
  • 00:12:00
    be some heaters in the flow field the
  • 00:12:02
    density of the parcel may change as the
  • 00:12:05
    fluid thermally expands when it's heated
  • 00:12:07
    so it's density and its volume may
  • 00:12:09
    change but we're still considering that
  • 00:12:11
    same mass or the same group of fluid
  • 00:12:14
    molecules that are moving through the
  • 00:12:16
    domain so its mass doesn't change we're
  • 00:12:18
    not adding or subtracting any molecules
  • 00:12:20
    from our parcel
  • 00:12:22
    and what that means is that we can
  • 00:12:23
    simplify our equation and we can take
  • 00:12:26
    the mass outside of the lagrangian
  • 00:12:29
    derivative term
  • 00:12:30
    and that allows us to arrive at equation
  • 00:12:32
    six
  • 00:12:34
    and this is a very important equation
  • 00:12:37
    and what we could do of course is we
  • 00:12:40
    could solve that equation directly we
  • 00:12:43
    could integrate the equation in time and
  • 00:12:45
    the solution of that equation which we
  • 00:12:48
    could calculate with some formula like
  • 00:12:50
    this if we were using an explicit Euler
  • 00:12:53
    method for the derivative it would tell
  • 00:12:55
    us how the velocity of that parcel of
  • 00:12:58
    fluid molecules varies in time as it
  • 00:13:01
    moves along its trajectory
  • 00:13:03
    and integrating that equation directly
  • 00:13:06
    is actually the approach that's used in
  • 00:13:08
    lagrangian particle tracking so all it's
  • 00:13:11
    telling us is how the velocity of that
  • 00:13:12
    parcel varies in time but that's not
  • 00:13:15
    actually what we want to do here we
  • 00:13:17
    don't want to know how the velocity of
  • 00:13:20
    that parcel of fluid varies in time we
  • 00:13:22
    want the navier Stokes equations that
  • 00:13:24
    are going to be applicable to the entire
  • 00:13:26
    fluid domain so we want the variation in
  • 00:13:29
    space as well as time so we're going to
  • 00:13:32
    have to do a little bit more work
  • 00:13:33
    but I just wanted to make you aware that
  • 00:13:35
    you could integrate this equation
  • 00:13:37
    directly and if you did just integrating
  • 00:13:39
    in time that would give you the
  • 00:13:41
    lagrangian particle tracking solution
  • 00:13:44
    of course we want the solution on a
  • 00:13:46
    fixed mesh and the solution on a fixed
  • 00:13:49
    mesh of course is where we take our
  • 00:13:52
    fluid domain and we divide it up into
  • 00:13:54
    fixed volumes of a certain size and
  • 00:13:57
    fluid Moves In and Out of the volumes
  • 00:13:59
    across the faces and these volumes of
  • 00:14:02
    course these fluid volumes have a
  • 00:14:04
    constant volume but they may have a
  • 00:14:06
    variable Mass so it's a different way
  • 00:14:08
    around to the lagrangian particle
  • 00:14:10
    tracking the density of the fluid coming
  • 00:14:12
    in and out may change and so the mass
  • 00:14:14
    may change in time as well
  • 00:14:18
    now
  • 00:14:19
    what we want to do is ultimately we need
  • 00:14:22
    to change between the lagrangian form
  • 00:14:25
    and this form on a fixed mesh and it
  • 00:14:28
    turns out of course the navier Stokes
  • 00:14:30
    equations are the same and they're
  • 00:14:32
    identical no matter which derivation you
  • 00:14:35
    use whether you use the lagrangian
  • 00:14:37
    derivation or if you just do the
  • 00:14:39
    standard derivation of the naviostopes
  • 00:14:41
    equations considering a fixed mesh where
  • 00:14:43
    you consider the change in the velocity
  • 00:14:45
    between the faces and shrink the
  • 00:14:47
    incremental volume to zero both of these
  • 00:14:50
    are going to lead to the same form of
  • 00:14:51
    the navi Stokes equations and that's
  • 00:14:53
    what I'm going to show but for now we're
  • 00:14:55
    going to carry on with this lagrangian
  • 00:14:56
    form and then we're going to shift back
  • 00:14:59
    to the form for a fixed mesh later on
  • 00:15:03
    and the first thing we're going to do
  • 00:15:04
    pushing on is we're going to note that
  • 00:15:07
    the navier Stokes equations in their
  • 00:15:09
    traditional form are written per unit
  • 00:15:10
    volume they all have all the terms have
  • 00:15:13
    units of force per unit volume so what
  • 00:15:15
    we're going to do is divide both sides
  • 00:15:16
    of our equations by the volume of this
  • 00:15:18
    cell
  • 00:15:19
    and ultimately this means that when we
  • 00:15:21
    have the full form of the navi Stokes
  • 00:15:23
    equations we can integrate it over the
  • 00:15:25
    volume of different cells and that will
  • 00:15:27
    allow us to write our equations in a
  • 00:15:30
    discrete algebraic form that can be
  • 00:15:32
    solved by a computer
  • 00:15:33
    so we've divided both sides by the
  • 00:15:35
    volume of the cell
  • 00:15:37
    and the next thing that's commonly done
  • 00:15:39
    in the derivation of the Navigator
  • 00:15:40
    Stokes equations is to separate the
  • 00:15:43
    force acting on the volume into two
  • 00:15:45
    different contributions the first
  • 00:15:48
    contribution are the forces or the
  • 00:15:50
    stresses acting on the surface of the
  • 00:15:53
    parcel I've shown these in red and then
  • 00:15:55
    the second contribution are the
  • 00:15:57
    components acting on the body or acting
  • 00:15:59
    on the volume and you can think of the
  • 00:16:01
    components acting on the volume as
  • 00:16:03
    forces like gravity which acts on the
  • 00:16:05
    physical volume of the body and then the
  • 00:16:07
    surface forces will come from things
  • 00:16:09
    like pressure and shear stress acting on
  • 00:16:12
    the surface and it's common in the
  • 00:16:14
    derivation of the naryostokes equations
  • 00:16:16
    to separate the net force acting on the
  • 00:16:19
    parcel into these two different
  • 00:16:20
    contributions
  • 00:16:22
    and how the contributions normally work
  • 00:16:24
    we normally have the Divergence of the
  • 00:16:26
    stress tensor that's the first term
  • 00:16:28
    which represents the surface forces
  • 00:16:31
    acting on the surface of the body and
  • 00:16:33
    then I'm using a lowercase f to denote
  • 00:16:36
    all the body forces from gravity and
  • 00:16:38
    other forces
  • 00:16:40
    and in this form you may have seen this
  • 00:16:43
    form of the navier Stokes equations
  • 00:16:45
    before and technically this is the
  • 00:16:48
    cauchy form of the momentum equations
  • 00:16:50
    it's not the full navier Stokes
  • 00:16:53
    equations yet and in order to arrive at
  • 00:16:55
    the full manostokes equations we'd need
  • 00:16:57
    to use a constitutive relationship for
  • 00:16:59
    the shear stresses for the stresses and
  • 00:17:01
    separate them out into pressure and
  • 00:17:03
    shear stress contributions but I'm not
  • 00:17:06
    going to do that here today I'm going to
  • 00:17:08
    move on from this form and focus on the
  • 00:17:10
    other terms in the equation
  • 00:17:13
    so what we can do now is we've actually
  • 00:17:16
    got the final form of the equation we
  • 00:17:19
    need but this is valid for the moving
  • 00:17:22
    parcel so we're moving with the parcel
  • 00:17:24
    if we integrate this equation directly
  • 00:17:27
    that will give us the velocity of the
  • 00:17:29
    parcel itself as we move through the
  • 00:17:32
    domain but what we want to do is switch
  • 00:17:34
    from that lagrangian description to an
  • 00:17:37
    eulerian description or a description
  • 00:17:39
    where we have the fixed volume and fluid
  • 00:17:41
    flows through it and it turns out we can
  • 00:17:44
    do that easily by just using the
  • 00:17:46
    definition of the material derivative
  • 00:17:48
    and expanding that term on the left hand
  • 00:17:51
    side d u d t
  • 00:17:53
    and what you can see now is that by
  • 00:17:55
    expanding the material derivative we've
  • 00:17:58
    got the variation of the background flow
  • 00:18:00
    field in time that's that first
  • 00:18:02
    contribution and we've also got the
  • 00:18:05
    advective contribution so as the
  • 00:18:07
    velocity of the fluid is moving momentum
  • 00:18:10
    through the domain that's this second
  • 00:18:12
    term the advection or convection term
  • 00:18:14
    and this varies in space so if we were
  • 00:18:18
    to solve this form of the equation we'd
  • 00:18:21
    have to integrate in space as well as
  • 00:18:23
    time and that's why this form of the
  • 00:18:27
    navi Stokes equations would will allow
  • 00:18:30
    us to calculate the variation of the
  • 00:18:32
    velocity and momentum in the entire
  • 00:18:35
    fluid domain and this is actually what
  • 00:18:37
    we want we're going to be integrating in
  • 00:18:39
    space as well as in time
  • 00:18:42
    and what I really want you to notice
  • 00:18:44
    from this is that actually this form of
  • 00:18:47
    the navierstopes equations is identical
  • 00:18:49
    to this form of the navier Stokes
  • 00:18:52
    equations if the equations are identical
  • 00:18:54
    The navier Stokes equations is uniformly
  • 00:18:56
    the same regardless of how we choose to
  • 00:18:58
    write it but the difference between the
  • 00:19:00
    lagrangian and the eulerian description
  • 00:19:03
    is how we choose to solve it in the
  • 00:19:05
    lagrangian form we just integrate this
  • 00:19:07
    directly whereas in the eulerian form we
  • 00:19:10
    actually expand and then we consider the
  • 00:19:13
    space and the time variations when we
  • 00:19:15
    solve so those forms of the navi Stokes
  • 00:19:18
    equations are identical and the material
  • 00:19:20
    derivative is a very convenient way of
  • 00:19:23
    getting to the navi Stokes equations
  • 00:19:25
    quickly without having to consider
  • 00:19:27
    incremental volumes and variations over
  • 00:19:29
    faces and shrinking the volume down to
  • 00:19:31
    an infinitesimal volume it's a very
  • 00:19:33
    convenient and easy derivation there
  • 00:19:37
    but what I'm going to move on to now is
  • 00:19:39
    show you two further forms of the navier
  • 00:19:41
    Stokes equations which are equivalent to
  • 00:19:44
    the the form that we've seen before and
  • 00:19:46
    you'll also see these cropping up in the
  • 00:19:48
    literature when you look at the
  • 00:19:49
    different forms of the navier Stokes
  • 00:19:51
    equations but it's important to remember
  • 00:19:52
    that all of these forms are identical
  • 00:19:55
    and that's the key takeaway from this
  • 00:19:57
    talk the form that you choose to use
  • 00:19:59
    depends on the form which is most useful
  • 00:20:01
    to you and what you're trying to do
  • 00:20:04
    and to show you the difference between
  • 00:20:06
    the advective form and the conservative
  • 00:20:08
    form we're going to be looking at the
  • 00:20:10
    left hand side of the equation here the
  • 00:20:13
    terms that are underlined with the
  • 00:20:15
    underbrace there and because I'm only
  • 00:20:17
    going to be looking at the left hand
  • 00:20:18
    side of the equation for the rest of
  • 00:20:20
    this talk we don't really need to
  • 00:20:22
    consider the right hand side anymore and
  • 00:20:24
    so I'm just going to combine those
  • 00:20:25
    together into the net external force
  • 00:20:27
    factor F over V and for you following
  • 00:20:30
    along if you're writing the equations
  • 00:20:32
    down and trying this for yourself you
  • 00:20:34
    can go with either of these forms or you
  • 00:20:36
    can even use a constitutive relationship
  • 00:20:39
    for the shear stress if you want and
  • 00:20:40
    write the full navier Stokes equations
  • 00:20:43
    on the right hand side with the shear
  • 00:20:45
    stress and the pressure if you want
  • 00:20:47
    the analysis will be the same I'm just
  • 00:20:49
    going to be using this compact form
  • 00:20:51
    because it makes the terms easier to
  • 00:20:53
    manage and navigate
  • 00:20:55
    now let's look at the advective form and
  • 00:20:58
    the conservative form
  • 00:21:00
    equation 11 is what we've been using so
  • 00:21:03
    far this is the form that arises when we
  • 00:21:05
    take the derivation of the naviest Oaks
  • 00:21:07
    equations regardless of if we use a
  • 00:21:09
    lagrangian type derivation or if we use
  • 00:21:12
    an eulerian type derivation which I
  • 00:21:14
    haven't used here we arrive at this form
  • 00:21:16
    of the navi Stokes equations and this is
  • 00:21:18
    commonly called the advective form or
  • 00:21:20
    the convective form of the navi Stokes
  • 00:21:22
    equations and the reason for that is
  • 00:21:24
    this term here represents the advection
  • 00:21:27
    of or the movement of momentum through
  • 00:21:30
    the domain by the velocity field itself
  • 00:21:33
    so this is representing momentum
  • 00:21:36
    but what we would like to do is rewrite
  • 00:21:38
    this equation in conservative form and
  • 00:21:42
    conservative form is the form you can
  • 00:21:43
    see there in equation 12. and what
  • 00:21:46
    differences do you notice well in the
  • 00:21:49
    conservative form all of the variables
  • 00:21:51
    so rho U and rho uu they appear inside
  • 00:21:55
    the operator whether that be the time
  • 00:21:58
    derivative or the Divergence operator
  • 00:22:00
    these operators operate on all of the
  • 00:22:03
    variables you can see there we don't
  • 00:22:05
    have any variables outside being
  • 00:22:07
    multiplied we don't have a density
  • 00:22:09
    outside and we don't have a velocity
  • 00:22:10
    outside either
  • 00:22:13
    so all of the variables are written in
  • 00:22:15
    conservative form and the reason that we
  • 00:22:18
    want to do that is it actually makes
  • 00:22:20
    things a lot easier when we apply the
  • 00:22:22
    finite volume method because we can
  • 00:22:24
    apply the Divergence Theorem to these
  • 00:22:26
    terms but I'm not going to be going into
  • 00:22:28
    that in this lecture in this lecture all
  • 00:22:30
    I want you to do is appreciate the
  • 00:22:32
    difference between the advective form of
  • 00:22:34
    the left hand side and the conservative
  • 00:22:37
    form of the right hand side of the left
  • 00:22:38
    hand side they are slightly different
  • 00:22:41
    but of course as we've seen so far in
  • 00:22:44
    this talk these forms are equivalent and
  • 00:22:47
    we can use either
  • 00:22:48
    but how can we show that this new
  • 00:22:51
    conservative form is actually equivalent
  • 00:22:53
    to the advective form that we've been
  • 00:22:55
    using so far
  • 00:22:56
    we're going to have to do a derivation
  • 00:23:00
    and if you've attempted this derivation
  • 00:23:02
    before or if you've searched for it on
  • 00:23:04
    the internet the key to the derivation
  • 00:23:06
    to showing that the advective and
  • 00:23:08
    conservative forms are the same is to
  • 00:23:10
    actually start with the conservative
  • 00:23:12
    form and work backwards it's a lot
  • 00:23:14
    easier to do it that way
  • 00:23:15
    and what I'm going to do in this talk is
  • 00:23:18
    to again start with the conservative
  • 00:23:20
    form and work backwards but I'm only
  • 00:23:22
    going to take the X component of the
  • 00:23:24
    navi Stokes equations so lowercase U I'm
  • 00:23:27
    going to be using rather than uppercase
  • 00:23:28
    U and I'm only going to be doing it in
  • 00:23:31
    2D so considering the X and the Y
  • 00:23:33
    components of the equation and if you
  • 00:23:36
    want to do this yourself you can
  • 00:23:37
    consider all three components and you
  • 00:23:39
    can do in 3D if you want but for showing
  • 00:23:41
    the equivalent you only really need to
  • 00:23:43
    do it in 2D with the X components of the
  • 00:23:45
    navi Stokes equations
  • 00:23:47
    so the X component is equation 14.
  • 00:23:50
    notice that we're using lowercase U for
  • 00:23:53
    the X component of the Velocity field
  • 00:23:54
    and in the uh the Divergence term here
  • 00:23:58
    the first velocity U is lowercase
  • 00:24:01
    because we're considering the U momentum
  • 00:24:03
    equation but of course to evaluate the
  • 00:24:06
    Divergence nabla dot this needs to be a
  • 00:24:08
    vector quantity which is why we've still
  • 00:24:11
    got the velocity Vector here but the
  • 00:24:14
    term that's being moved by the flow is
  • 00:24:17
    the U component of the momentum and the
  • 00:24:19
    force again lowercase f we're only
  • 00:24:22
    considering the X component of the force
  • 00:24:24
    Vector because this is the momentum
  • 00:24:26
    balance or conservation of momentum in
  • 00:24:29
    the X Direction only
  • 00:24:31
    and for the derivation how do we do it
  • 00:24:33
    the easy way to do it is to expand the
  • 00:24:36
    Divergence operator so replacing nablo
  • 00:24:39
    with d by DX and D by d y and when you
  • 00:24:42
    expand the terms out you see you've got
  • 00:24:43
    d by DX of rho uu and D by d y of rho UV
  • 00:24:47
    so these are the two terms
  • 00:24:50
    and then what we can do is use the
  • 00:24:52
    product rule and remember that the
  • 00:24:54
    product rule in mathematics for example
  • 00:24:56
    if we take this first term the rate of
  • 00:24:59
    change of row u in time that becomes
  • 00:25:01
    equal to rho d u d t plus u d rho DT
  • 00:25:05
    that's the product rule when we're
  • 00:25:07
    taking the derivative of two variables
  • 00:25:09
    multiplied together
  • 00:25:10
    and we can also apply the product rule
  • 00:25:12
    to the D by DX term which gives us rho u
  • 00:25:15
    d u d x plus u d rho u by DX and the
  • 00:25:19
    same for the Y term as well
  • 00:25:21
    and then what we're going to do is
  • 00:25:23
    collect all of the common terms together
  • 00:25:25
    so all of the terms that are multiplied
  • 00:25:28
    by rho you can see we've got one here
  • 00:25:30
    rho Duda DT and then rho u d u d x and
  • 00:25:34
    then another one here rho v d u d y and
  • 00:25:37
    collect those together in this first
  • 00:25:38
    bracket and then the second bracket I'm
  • 00:25:41
    going to collect all the terms together
  • 00:25:42
    that are multiplied by lowercase U so I
  • 00:25:45
    can see I've got a D rho DT here a d rho
  • 00:25:48
    u by DX here and a d rho v by d y there
  • 00:25:51
    so I've used the product rule and then
  • 00:25:53
    collected all the terms together and if
  • 00:25:55
    you need to write these down for your
  • 00:25:57
    for yourself at home go ahead and do
  • 00:25:58
    that it will help with your
  • 00:26:00
    understanding
  • 00:26:01
    and then what I'm going to do is
  • 00:26:02
    reintroduce nabla so reintroduce the
  • 00:26:05
    gradient operator and just looking at
  • 00:26:08
    these you can see where the gradient
  • 00:26:09
    operator is going to come in we've got a
  • 00:26:11
    d by DX here a d by d y so it's going to
  • 00:26:14
    be a gradient operator here and we've
  • 00:26:16
    got a d by DX here and a d by d y here
  • 00:26:19
    but you'll notice that the gradient
  • 00:26:20
    operator here is applied to row u and
  • 00:26:22
    rho v whereas over here it's applied to
  • 00:26:25
    U
  • 00:26:26
    so what does that mean
  • 00:26:28
    when we reintroduce the gradient
  • 00:26:29
    operator we arrive at this equation
  • 00:26:31
    equation 19 and you can see
  • 00:26:34
    the first bracket is still here
  • 00:26:36
    pre-multiplied by row and the second
  • 00:26:38
    bracket is here pre-multiplied by U
  • 00:26:41
    and the key to this derivation to
  • 00:26:43
    showing that the advective form and the
  • 00:26:45
    conservative form are equivalent is to
  • 00:26:47
    recall that the continuity equation so
  • 00:26:49
    conservation of mass is given by
  • 00:26:52
    equation 20 and this is valid for
  • 00:26:54
    compressible and incompressible flows
  • 00:26:57
    and you'll notice looking back at
  • 00:26:59
    equation 19 that this second bracket is
  • 00:27:02
    actually equal to the continuity
  • 00:27:03
    equation so this entire second bracket
  • 00:27:06
    is actually equal to zero we get rid of
  • 00:27:08
    it completely and that allows us to
  • 00:27:11
    arrive at just the first bracket which
  • 00:27:13
    is the advective form of the navi Stokes
  • 00:27:16
    equations so we started with this term
  • 00:27:19
    in the Box here this was the convective
  • 00:27:22
    form uh sorry the conservative form and
  • 00:27:25
    we've shown that that's equal to the
  • 00:27:26
    advective form
  • 00:27:28
    so both the conservative form and the
  • 00:27:31
    advective form and the lagrangian form
  • 00:27:35
    of the navi Stokes equations are all
  • 00:27:38
    equivalent all of these equations are
  • 00:27:40
    the same form of the navier Stokes
  • 00:27:43
    equations but we've just Rewritten them
  • 00:27:45
    slightly by using different operators
  • 00:27:47
    and rearranging the terms you can use if
  • 00:27:50
    you write your own if you write your own
  • 00:27:53
    papers and manuscripts and you're citing
  • 00:27:56
    The navier Stokes equations any three of
  • 00:27:58
    these forms of the left hand side of the
  • 00:28:00
    navi Stokes equations they're all
  • 00:28:02
    equivalent they all mean the same thing
  • 00:28:04
    but the difference is what we choose to
  • 00:28:07
    use them for and if we're going to be
  • 00:28:10
    doing lagrangian particle tracking then
  • 00:28:12
    we would probably take this form of the
  • 00:28:14
    equation and integrate it directly in
  • 00:28:17
    time
  • 00:28:18
    now if we're using the finite volume
  • 00:28:20
    method to solve the navi Stokes
  • 00:28:22
    equations then we would take the
  • 00:28:24
    conservative form which is at the bottom
  • 00:28:26
    the reason that we do that is it's
  • 00:28:28
    easier to apply the Divergence Theorem
  • 00:28:30
    in the finite volume method to this to
  • 00:28:33
    this form because all of the terms in
  • 00:28:35
    the equation are inside or being
  • 00:28:38
    operated on their respective derivatives
  • 00:28:42
    so those are the different forms of the
  • 00:28:44
    navier Stokes equations hopefully now
  • 00:28:47
    you can see that these forms are unified
  • 00:28:48
    and all represent the same equation it
  • 00:28:51
    only depends on what you choose to do
  • 00:28:53
    with them and actually it turns out
  • 00:28:55
    these two different forms the
  • 00:28:57
    conservative form and the objective form
  • 00:28:59
    also appear in the other transport
  • 00:29:01
    equations as well
  • 00:29:03
    and if you think about for example the
  • 00:29:05
    the equation for enthalpy
  • 00:29:08
    um the enthalpy equation may look
  • 00:29:10
    something like this equation 26 written
  • 00:29:13
    in advective form once again you can see
  • 00:29:15
    you've got rho multiplied by DH DT plus
  • 00:29:19
    u dot nabla h this is an adjective form
  • 00:29:23
    and that's going to be equal to we have
  • 00:29:26
    the conduction or diffusion term on the
  • 00:29:28
    right hand side and sources of enthalpy
  • 00:29:30
    as well and that form is of course
  • 00:29:33
    equivalent to a conservative form where
  • 00:29:36
    we could rewrite it with the rate of
  • 00:29:38
    change in time of rho h plus nabla dot
  • 00:29:41
    rho uh there as well so you can quite
  • 00:29:44
    clearly see that for these transport
  • 00:29:45
    equations we can write them in advective
  • 00:29:47
    form or conservative form they both
  • 00:29:50
    represent the same equation and actually
  • 00:29:52
    for the enthalpy equation you could go
  • 00:29:55
    through the same derivation that I just
  • 00:29:56
    went through for the navi Stokes
  • 00:29:58
    equations and you have to use all the
  • 00:30:00
    same techniques so start with the
  • 00:30:02
    convective form work backwards use the
  • 00:30:04
    product rule and then use the continuity
  • 00:30:06
    equation to cancel out some of those
  • 00:30:08
    terms and you can show that the
  • 00:30:10
    objective form and the conservative form
  • 00:30:12
    are the same but hopefully taking this
  • 00:30:14
    forward you can see that actually when
  • 00:30:17
    cfd codes in the cfd user manuals for
  • 00:30:20
    example present the different forms of
  • 00:30:22
    the navier Stokes equations they are
  • 00:30:24
    equivalent but the conservative form is
  • 00:30:26
    more useful because it allows us to
  • 00:30:28
    apply the Divergence Theorem in the
  • 00:30:30
    finite volume method
  • 00:30:33
    so just a quick summary to wrap up
  • 00:30:34
    everything I've talked about today the
  • 00:30:36
    material derivative or lagrangian
  • 00:30:38
    derivative which is capital of d by DT
  • 00:30:41
    you can think of that as the measured
  • 00:30:44
    rate of change moving with a sensor so
  • 00:30:47
    you're moving with a parcel of fluid or
  • 00:30:49
    a thermometer in a swimming pool and
  • 00:30:51
    you're looking at how the temperature
  • 00:30:52
    changes with time on a screen
  • 00:30:55
    that's what this material derivative
  • 00:30:57
    represents it's really useful for us
  • 00:30:59
    because we can use it to do a quick
  • 00:31:01
    derivation of the navierstopes equations
  • 00:31:04
    which is a lot more compact than a
  • 00:31:07
    derivation in eulerian form we have to
  • 00:31:09
    consider the different surfaces of an
  • 00:31:11
    infinitesimal volume
  • 00:31:13
    and we can convert readily from the
  • 00:31:15
    lagrangian to look to the eulerian form
  • 00:31:17
    of the equation just by expanding the
  • 00:31:20
    definition of the material derivative
  • 00:31:21
    and we can do that for the navierstopes
  • 00:31:23
    equations or for any other transport
  • 00:31:26
    equation as well the equations
  • 00:31:27
    themselves are the same and represent
  • 00:31:30
    the same conservation properties and
  • 00:31:34
    those are some very useful different
  • 00:31:36
    forms which you can use to represent it
  • 00:31:41
    so that brings me to the end of the talk
  • 00:31:43
    I'm really hoping at the end of this
  • 00:31:45
    talk that you're clear on the
  • 00:31:47
    differences between the different forms
  • 00:31:49
    of the navier Stokes equations and other
  • 00:31:51
    transport equations and you can use this
  • 00:31:54
    in your own work when you're reciting or
  • 00:31:56
    recalling the equations and have the
  • 00:31:58
    confidence that you know the differences
  • 00:32:00
    between the different forms of the
  • 00:32:02
    equations
  • 00:32:03
    if you found this talk useful let me
  • 00:32:05
    know in the comments section and let me
  • 00:32:06
    know if there are any other parts of the
  • 00:32:08
    notation or equations that are commonly
  • 00:32:10
    used in cfd that you'd like to see
  • 00:32:12
    explained in more detail
  • 00:32:15
    and thank you all very much for watching
  • 00:32:17
    and I'll see you in the next video
タグ
  • Navier-Stokes
  • transport equations
  • material derivative
  • Lagrangian
  • Eulerian
  • advective form
  • conservative form
  • CFD
  • temperature gradient
  • continuity equation