CFD - Behavior of PDEs Flashcards
Covers chapter 3 of John Anderson's Introduction to CFD book.
If the eigenvalues are all real what is the classification of the PDE?
Hyperbolic
Anderson CFD, Pg. 103
If the eigenvalues for a given PDE of arbitrary order are all complex, what is the classification of the PDE?
Elliptic
Anderson CFD, Pg. 103
When dealing with a hyperbolic equation of the second order, how many real characteristic curves are there in the XY space?
2
Anderson CFD, Pg. 104
What numerical schemes is typically employed to solve hyperbolic equations?
Marching Techniques
Anderson CFD, Pg. 107
Steady, inviscid supersonic flows are described by what type of PDE?
Hyperbolic
Anderson CFD, Pg. 107
Unsteady inviscid flows are described by what type of PDE?
Hyperbolic.
Anderson CFD, Pg. 109
For a parabolic equation of second order, how many characteristic directions run through a given point P?
One
Anderson CFD, Pg. 111
Parabolic equaitons lend themselves to ________ solutions.
Marching
Anderson CFD, Pg. 112
Steady boundary layer flows are modeled by what type of PDE?
Parabolic PDEs
Anderson CFD, Pg. 113
Unsteady thermal conduction is modeled by what type of PDE?
Parabolic PDEs
Anderson CFD, Pg. 115
Parabolic equations have how many characteristic lines?
One
Anderson CFD, Pg. 111
The characteristic curves for an elliptic equation are _________.
Imaginary
Anderson CFD, Pg. 117
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.
True
Anderson CFD, Pg. 117
Give some examples of flows that are governed by elliptic equations.
- Steady, subsonic, inviscid flows
- Incompressible inviscid flow
Anderson CFD, Pg. 118
True or False
In a theoretically incompressible flow, the Mach number is infinity.
True
Anderson CFD, Pg. 118
True or False
Unsteady inviscid flow is governed by hyperbolic equations no matter whether the flow is locally subsonic or supersonic.
True.
Anderson CFD, Pg. 119
Define a “well-posed” problem with respect to CFD.
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
Define a Linear PDE.
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
What is a non-linear PDE?
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
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?
(dy/dx) = (B +/- sqrt(B^2 - 4AC)) / (2A)
Note that we only care about the three highest-order derivatives.
Hoffmann’s CFD, Pg. 5
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?
Elliptic: dy/dx < 0
Parabolic: dy/dx = 0
Hyperbolic: dy/dx > 0
Hoffmann’s CFD, Pg. 5
True or False
An elliptic PDE has no real characteristic curves.
True.
Hoffmann’s CFD, Pg. 6
True or False
For a parabolic PDE, there exists one real characteristic line.
True.
Hoffmann’s CFD, Pg. 6
True or False
A hyperbolic PDE has two real characteristic lines.
True.
Hoffmann’s CFD, Pg. 8
Define “initial condition” with respect to ODEs.
An initial condition is a requirement for which the dependent variable is specified at some initial state.
Hoffmann’s CFD, Pg. 20
Define “boundary condition” with respect to PDEs.
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
What is the Dirichlet boundary condition?
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
What is a Neumann boundary condition?
If the normal gradient of the dependent variable along the boundary is specified.
Hoffmann’s CFD, Pg. 20
What is a Robin boundary condition?
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
