Finite Volume Methods

Question 1

What is the SIMPLE Algorithm?

Answer 1

• It obtains a velocity field by solving the momentum equations ( Doesnt satisfy continuity eq.)
• Solve Possion eq to get the pressure field
• Use the pressure field to correct velocity field in contiuity eq
• new velocity field doesnt satisfy momentum eq -> repeat cycle

Question 2

What is the purpose of finite volume methods?

Answer 2

Subdivision of domain into discrete elements

Question 3

What is the purpose of Spatial Discretization

Answer 3

Approximation of Spatial Gradients

Question 4

What is the purpose of Temporal Discretization?

Answer 4

Evolution of physical state variables in time

Question 5

How do Finite Volume Methods Work?

Answer 5

• Equations Discretized using a generic 3D polyhedral cell
• Flow variables calculated at cell centroid
• Incorporate neighbor cells
• Integrate Navier Stokes Equations across the cell
• Use different approach based on Complexity of Equation

Question 6

What is implicit treatment?

Answer 6

Adding linear source term to M matrix

Question 7

What is explicit treatment?

Answer 7

Adding linear source term to b vector

Question 8

How do you deal with the convection term in FVM?

Answer 8

Use divergence theorem or Gauss and convert volume integral to surface integral

Question 9

What is non-orthogonality in CFD?

Answer 9

Shared faces have a face angle

Question 10

Why are the Naver Stokes Equations difficult to solve numerically?

Answer 10

1. No equation for p, only for velocity components
2. The computed velocity fields from the momentum equations must satisfy the continuity equations
3. The convection term in the momentum equation is non linear
4. We cant use equation of state to compute pressure because density and temperature may be constant

Question 11

How do we solve the pressure velocity coupling problem for Steady incompressible Navier Stokes equations?

Answer 11

-SIMPLE Algorithm

• Convert the equations into matrix form Mu = - /nabla p where M are the coefficients, p is the pressure and u is the velocity

Question 12

What are five important terms in the momentum equation?

Answer 12

• Temporal derivative
• Convection
• Pressure gradient
• Diffusion term
• Source Term

Question 13

What is the formula for the matrix form of NSE?

Answer 13

Mu = -\nabla p

Question 14

How are constant source terms like gravity handled in the SIMPLE algorythm?

Answer 14

Append them to the b vector

Question 15

How do you solve the SIMPLE algorythm?

Answer 15

1. Solve momentum equation for velocity field
2. Solve Poisson equation for pressure field
3. Use pressure field to correct velocity field so it satisfies continuity equation
4. The velocity field now does not satisfy momentum equation so iterate