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Mukhopadhyay May 24, 2001 08:08

Driven cavity problem
In the (lid) driven cavity problem, we incorporate the lid velocity thru the Reynolds No. What happens to the Froude No. then? Or, we benchmark without the body force term (g=0), i.e., in a zero-gravity situation ? Will someone educate me please? Thanks in advance.

John C. Chien May 24, 2001 12:06

Re: Driven cavity problem
(1). It is a test case, a very good test case, designed to test the numerical algorithm with gravity not included.

Dr Strangelove May 24, 2001 15:39

Re: Driven cavity problem

For the classic [1] lid-driven cavity benchmark problem the flow is assumed to be incompressible, i.e.,
<blockquote> div V = 0. </blockquote>
In particular, solutions of the incompressible Navier-Stokes equations are simplified by the observation that the pressure may be written as the sum of a "dynamic pressure" and an additional pressure that, for lack of a better term, we will call "hydrostatic". What I call "dynamic" is referred to as the "modified pressure" by Batchelor[2], who explains this concept very well.

This "hydrostatic" component of the pressure may be used to match the gravity force term in the momentum equation. Indeed, any conservative force field may be eliminated from the equations in this manner, leaving no gravity term to complicate the situation. This is why, in practice, you never see authors including the gravity term in their presentations - it is immaterial (or should be).

If you like, you can always reincorporate the eliminated component of pressure at the end using analytic expressions to match whatever gravitational constant (or, equivalently) Froude number you desire.

However, you should notice from all this that the velocity field for this kind of problem remains independent of the gravity term. It is only the pressure that will change, and its change will be governed by the observation above.
<hr> [1] Burggraf, O. R., J Fluid Mech, 24, 113-151, 1966. [2] Batchelor, G. K., An Introduction to Fluid Dynamics, Cambridge University Press, Section 4.1, 1967.

Mukhopadhyay May 25, 2001 00:31

Re: Driven cavity problem
Thank you Dr. Chien and Dr.Strangelove.

* In such situation should we not revisit the 'Driven cavity flow - Benchmarking'?

* Should we not solve the 'full' NS Eqn without ignoring the body force term - rho*grav?

* Should we not benchmark the problem with Reynolds No and Froude No vis-a-vis the obtained (from numerical solution) u-v (non-dimensional) field AND the p (pressure - non-dim) field together?

* Then only, shall we not develop confidence w.r.t., the force balance and above all the mass conservation?

John C. Chien May 26, 2001 05:22

Re: Driven cavity problem
(1). The reason why the lid driven cavity flow was selected as a test case for Navier-Stokes equations was also because it was closely related to the flow in the journal bearing groove, or seals for turbomachinery. (2). In those cases, the dimension of the groove or the seal cavity is relatively small. Therefore, the effect of gravity is not included. (3). Any time, when you have a new algorithm to solve the Navier-Stokes equations, the lid driven cavity flow should be used to check out the performance (of the algorithm, not the cavity flow itself). This will eliminate un-necessary complication from the geometry.

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