Hello,

Is the presence of boussinessq approximations makes the NS momentum equation non-conservative?

Does it mean that a variable body force term makes the eqn non-conservative?

Thanks,

CFDtoy

In what sense are you using the term "non-conservative"? The momentum equations are simply a force balance, and represent a conservative system if all external forces are derivable from some potential functions. On the other hand, whether the system is a conservative one or not, the equations describe the "conservation of momentum", in which the time rate of change of momentum is balanced by external forces such as viscous shear and pressure acting over a surface area. We also can write the equations in a conservative or non-conservative form. Your question leaves a lot of room for interpretation.

Thanks ag. Ok, lets expand it then.

Writing conservative form of NS momentum eqn with a body force term, the bousinessq has force term F ~ function(Temperature). Now this term can vary in magnitude and can influence the U variable.

This force, bousinessq approximation, is it a non-conservative type?

if it is non-conservative force, we add this to the conservative Momentum eqn resulting in the overall eqn for the velocity variable.

After the addition of the non-conservative forces is the overall fluid momentum conserved? (My question is exactly based on your answer..since U eqn is primarily a momentum balance, adding a non-conserved variable..does it alter the conservation of momentum?)

Thanks,

CFDtoy

I don't know if I get this right. But why should the boussinesq-approximation be non-conservative? It just gives the relation between density and temperature. With this the bodyforce varies in magnitude, but the equation itself has not changed.

Forces in the momentum equation are generally non-conservative. If the force is actually there then it needs to be in the momentum equation (after all it is the sum of the external forces that give the change in momentum). Remember - conservation of momentum simply says that the sum of the external forces acting on a body is equal to the time reate of change of momentum. There's nothing in there about the nature of the individual forces.

Thats what I'm talking about.

Ag:

So, by adding a non-conservative force to the momentum eqn, the fluid momentum is conserved?

Although the momentum eqn is by itself a statement on F = ma type, and we are adding another force, it just to say,

F = ma+F2

so now, the U field has one more term to accommodate.

But is momentum conserved or maintained in the domain?

Thanks,

CFDtoy

Dear CFDtoy,

The conservation principle is stated in the = sign.

External forces + F2 = ma

regardless of what shows up on the left hand side, the equation says: that momentum is the summation of external forces and it does not say anything about the nature, or origin of this forces.

By the way, the gravity force can be written as the gradient of a potential.. That potential is the Potential energy. The fact that density is a function of temperature should be irrelevant. The force is conservative if it can be derived from a potential. Its closed contour (line) integral is 0, and satisfies several other theorems.

Not sure what you are after,

Opaque

Hello Opaque:

Are you saying the boussinessq approx is conservative?

CFDtoy

Hello Opaque:

Are you saying the boussinessq approx is conservative?

CFDtoy

CFDtoy,

I think you are confused due to the terms "conservative" and "conservation", which opaque tried to explain. Let me try to say it in other words.

A "conservative force" is defined as a force that may be expressed as the gradient of a scalar potential. For example, if the fluid is incompressible with density rho0 and gravity is acting in z direction, the potential is rho0*g*(z-z0), where z0 is some arbitrary reference z-location. The resulting (conservative) body force is therefore its gradient, i.e., the vector {0, 0, rho0*g}. On the other hand, the Boussinesq approximation cannot be derived from a potential, and is therefore a non-conservative body force.

The momentum equations are expressing the conservation of momentum, which is actually Newton's 2nd law. As such, it may include various body forces (conservative and non-conservative) while still maintaining equilibrium between the inertial and external contribution (i.e. conserving the momentum in vector form). Specifically, even if you use the (non-conservative) Boussinesq approximation as one of the body force terms, momentum is still conserved, as the momentum equations still obey Newton's 2nd law.

I hope this contributed to clear the misunderstanding of the terms.

Rami

"Specifically, even if you use the (non-conservative) Boussinesq approximation as one of the body force terms, momentum is still conserved, as the momentum equations still obey Newton's 2nd law."

Actually I think this is what's causing the confusion. Strictly speaking this is not conservation of momentum (which would require momentum to remain unchanged along a particle path) but an application of Newton's second law which states that the "rate of change of momentum = sum of forces acting". The Navier-Stokes equations are a statement of force balance rather than conservation - this is why they are usually refered to as "the momentum equations" or "the momentum balance" in Journals such as JFM.

In the theory of hyperbolic systems equations of the form dq/dt = div(F) are referred to as conservation laws. The adoption of this terminology within CFD can be confusing since the word is being used in two different contexts.