CFD - Behavior of PDEs

Question 1

If the eigenvalues are all real what is the classification of the PDE?

Answer 1

Hyperbolic

Anderson CFD, Pg. 103

Question 2

If the eigenvalues for a given PDE of arbitrary order are all complex, what is the classification of the PDE?

Answer 2

Elliptic

Anderson CFD, Pg. 103

Question 3

When dealing with a hyperbolic equation of the second order, how many real characteristic curves are there in the XY space?

Answer 3

2

Anderson CFD, Pg. 104

Question 4

What numerical schemes is typically employed to solve hyperbolic equations?

Answer 4

Marching Techniques

Anderson CFD, Pg. 107

Question 5

Steady, inviscid supersonic flows are described by what type of PDE?

Answer 5

Hyperbolic

Anderson CFD, Pg. 107

Question 6

Unsteady inviscid flows are described by what type of PDE?

Answer 6

Hyperbolic.

Anderson CFD, Pg. 109

Question 7

For a parabolic equation of second order, how many characteristic directions run through a given point P?

Answer 7

One

Anderson CFD, Pg. 111

Question 8

Parabolic equaitons lend themselves to ________ solutions.

Answer 8

Marching

Anderson CFD, Pg. 112

Question 9

Steady boundary layer flows are modeled by what type of PDE?

Answer 9

Parabolic PDEs

Anderson CFD, Pg. 113

Question 10

Unsteady thermal conduction is modeled by what type of PDE?

Answer 10

Parabolic PDEs

Anderson CFD, Pg. 115

Question 11

Parabolic equations have how many characteristic lines?

Answer 11

One

Anderson CFD, Pg. 111

Question 12

The characteristic curves for an elliptic equation are _________.

Answer 12

Imaginary

Anderson CFD, Pg. 117

Question 13

True or False
For elliptic equations there are no limited regions of influence or domains of dependence; rather, information is propagated everywhere and in all directions.

Answer 13

True

Anderson CFD, Pg. 117

Question 14

Give some examples of flows that are governed by elliptic equations.

Answer 14

1. Steady, subsonic, inviscid flows
2. Incompressible inviscid flow

Anderson CFD, Pg. 118

Question 15

True or False
In a theoretically incompressible flow, the Mach number is infinity.

Answer 15

True

Anderson CFD, Pg. 118

Question 16

True or False
Unsteady inviscid flow is governed by hyperbolic equations no matter whether the flow is locally subsonic or supersonic.

Answer 16

True.

Anderson CFD, Pg. 119

Question 17

Define a “well-posed” problem with respect to CFD.

Answer 17

If the solution to a partial differential equation exists and is unique and its solution depends continuously upon the initial and boundary conditions then the problem is well posed.

Anderson CFD, Pg. 120

Question 18

Define a Linear PDE.

Answer 18

In a linear PDE, there is no product of the dependent variable and/or its derivatives anywhere in the equation.

Hoffmann’s CFD, Pg. 3

Question 19

What is a non-linear PDE?

Answer 19

A PDE that contains the product of the dependent variable and/or a product of its derivatives somewhere in the equation.

Hoffmann’s CFD, Pg. 3

Question 20

For a general second-order PDE in the form
A(∂^2U/∂x^2) + B(∂^2U/∂x∂y) + C(∂^2U/∂y^2) = H
What is the formula for the equations of the characteristics?

Answer 20

(dy/dx) = (B +/- sqrt(B^2 - 4AC)) / (2A)

Note that we only care about the three highest-order derivatives.

Hoffmann’s CFD, Pg. 5

Question 21

In regards to the characteristic formula for determining the type of PDE (dy/dx = (B +/- sqrt(B^2 - 4AC) / (2A)), how is it determined if the equation is elliptic, parabolic, and hyperbolic respectively?

Answer 21

Elliptic: dy/dx < 0
Parabolic: dy/dx = 0
Hyperbolic: dy/dx > 0

Hoffmann’s CFD, Pg. 5

Question 22

True or False
An elliptic PDE has no real characteristic curves.

Answer 22

True.

Hoffmann’s CFD, Pg. 6

Question 23

True or False
For a parabolic PDE, there exists one real characteristic line.

Answer 23

True.

Hoffmann’s CFD, Pg. 6

Question 24

True or False
A hyperbolic PDE has two real characteristic lines.

Answer 24

True.

Hoffmann’s CFD, Pg. 8

Question 25

Define “initial condition” with respect to ODEs.

Answer 25

An initial condition is a requirement for which the dependent variable is specified at some initial state.

Hoffmann’s CFD, Pg. 20

Question 26

Define “boundary condition” with respect to PDEs.

Answer 26

A boundary condition is a requirement that the dependent variable or its derivative must satisfy on the boundary of the domain of the PDE.

Hoffmann’s CFD, Pg. 20

Question 27

What is the Dirichlet boundary condition?

Answer 27

If the dependent variable along the boundary is prescribed.

Example: Everywhere along a flat plate parallel to the flow, the temperature is 300 K.

Hoffmann’s CFD, Pg. 20

Question 28

What is a Neumann boundary condition?

Answer 28

If the normal gradient of the dependent variable along the boundary is specified.

Hoffmann’s CFD, Pg. 20

Question 29

What is a Robin boundary condition?

Answer 29

If the imposed boundary condition is a linear combination of the Dirichlet and Neumann types it is known as the Robin Type.

Hoffmann’s CFD, Pg. 20