Navier-Stokes Equations for a Compressible Fluid

[Physicist97]

Hello, I don't know if this question belonged here or in General Physics, so I apologize if I made a mistake. My question is simple, what are the Navier-Stokes Equations for a Compressible Fluid? I don't mean from a conceptual point of view, what I mean are the mathematical equations themselves. I've been looking left and right on the internet to find them, but every website gives them in different forms and I don't know which website is correct, or if all of them are correct but simply written differently. Thank you for any help.

 

[Chestermiller]

I think you will find what you are looking for in Transport Phenomena by Bird, Stewart, and Lightfoot.

Chet

 

[boneh3ad]

It really depends on which form you are using. The traditional version of the equations are not for compressible flows but there are versions that exist that are perfectly valid.

 

[Chestermiller]

Hello, I don't know if this question belonged here or in General Physics, so I apologize if I made a mistake. My question is simple, what are the Navier-Stokes Equations for a Compressible Fluid? I don't mean from a conceptual point of view, what I mean are the mathematical equations themselves. I've been looking left and right on the internet to find them, but every website gives them in different forms and I don't know which website is correct, or if all of them are correct but simply written differently. Thank you for any help.

Show us what forms of the equations you are referring to.

I hope you are aware that the NS equations are a combination of two other equations: (1) the general momentum balance equation, expressed in terms of the divergence of the stress tensor and (2) the rheological constitutive equation for a Newtonian fluid (including compressibility if required), expressed in terms of the isotropic pressure and the rate of deformation tensor. Which of these equations are you having difficulty with, if either? Or, is it just that you don't understand how the equations are combined?

Chet

 

[GiuseppeR7]

The NS stokes equations are in the classical sense the momentum equations applied to the continuum of the fluid but in the modern litterature NS equations are the system of PDE equations (including the momentum) describing the fluid (energy, continuity,...) with the proper constitutives relation.
The compressible NS equations are the NS equations without the Bousinneq simplification or anything like that.

 

[boneh3ad]

It has nothing to do with the Boussinesq approximation, which can mean one of a few things but typically involves assuming density fluctuations are small except when multiplied by gravity in buoyant flows or else leads to the concept of eddy viscosity in turbulence modeling.

In terms of the Navier-Stokes equations, whether or not they are valid for compressible flow or not depends on whether $\nabla \cdot \vec{v} = 0$ was assumed in the derivation. If not, there will be a second coefficient of viscosity and dilatational terms in the equations that otherwise would have been zeroed in the case of the more classical NS equations.

 

[GiuseppeR7]

is this what you said? or i am mistaken?

 

[boneh3ad]

I have literally never heard that referred to as the Boussinesq approximation in any of the perhaps dozen fluid mechanics books I own, and in general when you say Boussinesq approximation, that's not what people generally assume you mean. What's the source, here? Now you've got me curious.

 

[GiuseppeR7]

It' s the textbook that we use in our fluid mechanics course by Kundu and Cohen

 

[Chestermiller]

I have literally never heard that referred to as the Boussinesq approximation in any of the perhaps dozen fluid mechanics books I own, and in general when you say Boussinesq approximation, that's not what people generally assume you mean. What's the source, here? Now you've got me curious.

I've never heard of it referred to that way either. In any event, the continuity ratio is regarded as separate from the NS equations.

 

[GiuseppeR7]

I've never heard of it referred to that way either. In any event, the continuity ratio is regarded as separate from the NS equations.

yes that is true but Anderson is suggesting a change in the trend (even if he is really referring to CFD)

 

[Chestermiller]

yes that is true but Anderson is suggesting a change in the trend (even if he is really referring to CFD)

Interesting.

Chet

 

[boneh3ad]

I've heard a lot of people (including myself) use "Navier-Stokes" to refer to the whole system. I generally hear it (and use it) to refer to the full compressible equations (continuity, momentum, energy and state) and then try to specify incompressible or compressible to remove any ambiguity. Either way, if a $\lambda$ or $\nabla\cdot\vec{v}$ show up in the momentum equations, you can be pretty confident in assuming that the equations are at least intended for use in a compressible flow.

Also, I believe I see where this other definition of "Boussinesq approximation" is originating. Basically, the Boussinesq approximation says that changes in $\rho$ are too small to affect the flow field except when they arise in conjunction with gravity (i.e. they only matter when they give rise to buoyancy effects). One of the consequences of this is that the $D\rho/Dt$ term in the continuity equation is eliminated since it is assumed to be small, reducing the continuity equation to its incompressible form. However, this is not the same as simply assuming the flow is incompressible in the first place, even if they are functionally similar. For example, it makes no sense to make a Boussinesq approximation in a horizontal, isothermal flow of air at 20 m/s at sea level over a sphere. That is not a situation where buoyancy is likely to matter. However, it is perfectly justified to call that incompressible.

I'll also add that while the Boussinesq approximation results in the incompressible continuity equation, the momentum equations using the Boussinesq approximation are not identical to those obtained from the incompressible assumption. Using the Boussinesq approximation, the body force term does not assume a constant density like the convective term does, and so density is still a variable, not a constant.

 

[Andy Resnick]

Show us what forms of the equations you are referring to.

I hope you are aware that the NS equations are a combination of two other equations: (1) the general momentum balance equation, expressed in terms of the divergence of the stress tensor and (2) the rheological constitutive equation for a Newtonian fluid (including compressibility if required), expressed in terms of the isotropic pressure and the rate of deformation tensor. Which of these equations are you having difficulty with, if either? Or, is it just that you don't understand how the equations are combined?

Chet

I agree (Bird's Transport Phenomena is a great text, Slattery's "Interfacial Transport Phenomena" is my go-to reference), but to slightly expand, I would say there are 3 relevant equations (mass balance, momentum balance, and energy balance) and one or more constitutive relations to account for material properties. And to be really ugly, add in jump conditions as needed.

 

[Physicist97]

Hello guys, thanks for all the amazing replies! I looked into Transport Phenomena by Bird and found the equation written ${\rho}{\frac{Du}{Dt}}={\mu}{\nabla^{2}}u-{\nabla}p+f_{body}$ , where $f_{body}$ is a vector that indicates the body force density on the fluid and ${\mu}$ is a viscosity constant. In Transport Phenomena $f_{body}$ was simply taken as gravity. $u$ is also a vector (I'm sorry I don't know how to use LaTeX notation to put an arrow over the vector). I had also heard this equation stated as ${\rho}{\frac{Du}{Dt}}={\nabla}{\sigma}+f_{body}$ . Where ${\sigma}$ is the stress tensor. This is the equation I am confused about, unfortunately I can not find the reference where this equation was written. I would like to find the NS Equations written in a form using the stress tensor. I also now that the ${\nabla}({\rho}u)=0$ , but I don't know of this is a continuity equation or the equation for conservation of mass. Thank you for all the help, guys.

 

[boneh3ad]

"Conservation of mass" and "continuity equation" are the same thing. Additionally, the equation you have reproduced here is not valid for compressible flow.

 

[Chestermiller]

Hello guys, thanks for all the amazing replies! I looked into Transport Phenomena by Bird and found the equation written ${\rho}{\frac{Du}{Dt}}={\mu}{\nabla^{2}}u-{\nabla}p+f_{body}$ , where $f_{body}$ is a vector that indicates the body force density on the fluid and ${\mu}$ is a viscosity constant. In Transport Phenomena $f_{body}$ was simply taken as gravity. $u$ is also a vector (I'm sorry I don't know how to use LaTeX notation to put an arrow over the vector). I had also heard this equation stated as ${\rho}{\frac{Du}{Dt}}={\nabla}{\sigma}+f_{body}$ .

This is just the momentum balance equation, without being combined with the mechanical constitutive equation for a Newtonian fluid. It applies to any material, not just a fluid. Included here would be a solid, or a viscoelastic material, etc. Incidentally, there is supposed to be a "dot" between the ∇ operator and the stress tensor σ. It should read:
$${\rho}{\frac{D\vec{u}}{Dt}}=\vec{\nabla} \centerdot \vec{σ}+\vec{f}_{body}\tag{momentum balance eqn.}$$
So this is not the NS equation(s) until it is combined with the constitutive equation for a Newtonian fluid. For the special case of an incompressible Newtonian fluid, for example, that equation would read:
$$\vec{σ}=-p\vec{I}+μ\left(\vec{∇} \vec{u}+(\vec{∇} \vec{u})^T\right)$$
where $\vec{I}$ is the identity tensor and $(\vec{∇} \vec{u})^T$ is the transpose of the velocity gradient tensor $\vec{∇} \vec{u}$.

I would like to find the NS Equations written in a form using the stress tensor.

There is no such thing, because the NS Equations have already eliminated the stress tensor by combining the momentum balance equation with the constitutive equation for a Newtonian fluid. That being said, the equation you are really looking for is the momentum balance equation (above).

I also now that the ${\nabla}({\rho}u)=0$

This is the continuity equation for cases in which the fluid density is not changing with time (i.e., steady state flow). There should be a "dot" in this equation also:
$$\vec{\nabla} \centerdot (\rho \vec{u})=\vec{0}$$

I don't know if this is a continuity equation or the equation for conservation of mass.

The continuity equation and the equation for conservation of mass are synonymous.

Chet

 

[Physicist97]

"Conservation of mass" and "continuity equation" are the same thing. Additionally, the equation you have reproduced here is not valid for compressible flow.

What would the equation be for a compressible fluid? Also, I did not know that conservation of mass and the continuity equation were equivalent, thank you and Chestermiller for that information.

 

[Chestermiller]

What would the equation be for a compressible fluid? Also, I did not know that conservation of mass and the continuity equation were equivalent, thank you and Chestermiller for that information.

If you want the constitutive equation for a compressible Newtonian fluid, see Eqn. 1.2-7 in Bird, Stewart, and Lightfoot, in which, in our notation, $\vec{τ}=-(\vec{σ}+p\vec{I})$ and $\vec{δ}=\vec{I}$.

Chet

 

[boneh3ad]

If you want the constitutive equation for a compressible Newtonian fluid, see Eqn. 1.2-7 in Bird, Stewart, and Lightfoot, in which, in our notation, $\vec{τ}=-(\vec{σ}+p\vec{I})$ and $\vec{δ}=\vec{I}$.

Chet

Maybe I am missing something (literally, I suppose), but did you forget the dilatational term? There ought to be a $\lambda I \nabla\cdot\vec{v}$ somewhere, right?

 

[Chestermiller]

Maybe I am missing something (literally, I suppose), but did you forget the dilatational term? There ought to be a $\lambda I \nabla\cdot\vec{v}$ somewhere, right?

Yes, right on. The dilatational term is in the explicit expression that Bird et al give for $\vec{τ}$ (Eqn. 1.2-7).

Chet

 

[boneh3ad]

Yes, right on. The dilatational term is in the explicit expression that Bird et al give for $\vec{τ}$ (Eqn. 1.2-7).

Chet

As that is not in my library, I had no way of checking what was actually in Bird et al., and I am sure you are aware of how obnoxious it is that different authors use many different notations for these things.

At any rate, then, the stress tensor ought to be
$$\tau_{ij} = -p\delta_{ij} + \mu\left( \dfrac{\partial u_i}{\partial x_j} + \dfrac{\partial u_j}{\partial x_i} \right) + \delta_{ij}\lambda \dfrac{\partial u_k}{\partial x_k}.$$

For an incompressible flow, where $\nabla\cdot\vec{v} = \partial u_k/\partial x_k = 0$, this reduces to the more familiar stress tensor
$$\tau_{ij} = -p\delta_{ij} + \mu\left( \dfrac{\partial u_i}{\partial x_j} + \dfrac{\partial u_j}{\partial x_i} \right)$$
and $\lambda$ is not important.

I suppose I should also point out that since the interest here is compressible flow, you cannot ignore the energy equation. In an incompressible flow, the density is treated as constant and the energy equation can be solve after solving for the flowfield. This is not the case for a compressible flow. Since density is now a variable, you need an additional equation. The energy equation can cover the density variable, but also introduces temperature as a variable (which is, in general, not constant), so you need a sixth equation to solve the system. This is an equation of state. Typically, for a gas, this ends up being the ideal gas law (or perhaps a real gas law if you care about real gas effects).

The moral of the story is you need 6 equations now: mass, 3 momentum equations, energy, and an equation of state.

 

[Chestermiller]

As that is not in my library, I had no way of checking what was actually in Bird et al., and I am sure you are aware of how obnoxious it is that different authors use many different notations for these things.

At any rate, then, the stress tensor ought to be
$$\tau_{ij} = -p\delta_{ij} + \mu\left( \dfrac{\partial u_i}{\partial x_j} + \dfrac{\partial u_j}{\partial x_i} \right) + \delta_{ij}\lambda \dfrac{\partial u_k}{\partial x_k}.$$

Hi Boneh3ad,

I'm going to try to get us all on the same page with regard to the notational differences.

What you call τ_ij, the OP calls σ_ij and BSL calls -pδ_ij-τ_ij. Thus, the BSL τ_ij is not the same as your τ_ij, and they use the unusual sign convention that compressive stresses are positive.

What you call λ, BSL call $(k-\frac{2}{3}μ)$, where k is what they call the dilatational viscosity. For a monatomic gas, k = 0.

So, in terms of the tensor object notation that I used in an earlier post, for a compressible Newtonian fluid:
$$\vec{σ}=-p\vec{I}+μ\left(\vec{∇} \vec{u}+(\vec{∇} \vec{u})^T\right)+λ(\vec{∇}\centerdot \vec{v})\vec{I}=-p\vec{I}+μ\left(\vec{∇} \vec{u}+(\vec{∇} \vec{u})^T\right)+(k-\frac{2}{3}μ)(\vec{∇}\centerdot \vec{v})\vec{I}$$

Hope this helps.

Chet