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Discretization of Fluid Models(Navier Stokes)

Dr. Farzad Ismail

School of Aerospace and Mechanical EngineeringUniversiti Sains Malaysia

Nibong Tebal 14300 Pulau Pinang

Week 5 (Lecture 1 and 2)

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Preview

We have talked about various schemes to solve model problems.

Would like to use the knowledge to solve real fluid models.

Before we do that, first need to understand the mathematical and physical nature of fluid dynamics.

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The Compressible Navier Stokes

One of the most complete mathematical models for fluids.

Includes compressibility, viscous, heat transfer, advection, pressure effects.

Can also be used to account for reacting fluids

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2D Compressible NS

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The Compressible Navier Stokes (cont’d)

In 2D, a system of 4 x 4 (3D - 5 x 5)

A hybrid of hyperbolic and parabolic types for unsteady cases

Elliptic in nature for steady cases

Decompose NS model into inviscid (compressible) and viscous (incompressible) parts

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2D Incompressible Navier Stokes (NS)

How do you know that mass equation is numerically satisfied?

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Pressure Poisson

(1) (2) (3) (4)

(1)

(4)

(3)

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Pressure Poisson (cont’d)

(3)

Solve * and ** for incompressible NS

(*)

(**)

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2D Incompressible Navier Stokes (NS)

The momentum can be rewritten

(***)

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Exercise

Take the gradient of Eqn (***), apply themass equation and show that

What does this equation provide?

More importantly, what is the nature of this eqnAnd how to solve it?

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2D Incompressible NS (cont’d)

Incompressible flow has only mass and momentum equations– The energy equation drops out (3 eqns, 3 unknowns)

Mass is implicitly solved through the pressure-Poisson equation– Pressure-based solver (i.e. SIMPLE, PISO)– Requires iterations to satisfy velocity divergence- expensive!

Adds an elliptic nature to pde on top of the hyperbolic and parabolic natures

Flow is ‘smooth’

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Choice of Grid Discretization

It is not critical for unsteady problems but it is crucial for steady-state problems– Elliptic problems have issues with certain errors not being

removed even after many iterations.

Collocated: all variables are stored at the same location– Worst choice, since there will be some errors that will never be

removed!

Compact: – Better choice, but there will be at least one type of errors that will

never be removed!

Staggered- best choice for elliptic problem, but…

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Staggered Grid

What about p, F,G, H’s?

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Initial & Boundary Conditions

For elliptic problems, IC is not so much a big deal– Will just determine how quick solution reach steady-state

But the BC is very critical!– Not only determines the efficiency of the computations– But will also determine what is the final outcome– Unphysical BC may lead to huge problems?Why?

Concept of Ghost Cells

No slip condition along and no outflow/inflow through boundaries

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Boundary Conditions (cont’d)

How to compute ghost cells?

– Combine BC and the equations of motions to extrapolate the values outside the computational domain

– Take horizontal boundary, v=0 (obvious) but u may not be unless wall is not moving (no slip)

– Can use mass equation

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Boundary Conditions (cont’d)

Nature of u,v at boundaries

– v=0 at boundary but u may not be zero; v even function– vy = -ux = 0 (by mass), implies that u is constant along walland that integrating u wrt x gives u(x,y) as a constant plus a linearVariation along y. – The ghost quantities (previous Fig) via the y-momentum:

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Vo and H(y) are zero on the boundary, this simplifies to

where vT are expanded from v0 via Taylor series

Since v0=0 and assume that the square of v and the high order terms are negligible, hence

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Since v is an even function,

The same can be done for other walls and similar approach for (u,v)

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Algorithm for Incompressible NS

Apply the appropriate BC

Solve for the preliminary u*, v* based on (u,v,p) of IC and BC

Compute the Pressure-Poisson equation to solve for p* – Iterate until reach required error tolerance

Check the discrete velocity divergence, if criterion is met, proceed to the next time level. If not go back to step 1 and continue the iteration within the same time-level