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# Computational Fluid Dynamics – Mathematical Behaviour of PDE – Well Posed Problems

1 - Question

When can we say that a problem is suitable to be solved using CFD?

a) The PDE has no solution

b) The solution to PDE is unique and it depends continuously on the initial and boundary conditions

c) The PDE has more than one solution

d) The solution to PDE is unique and independent of the initial and boundary conditions

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Answer: bExplanation: We say a problem to be well suited for CFD when the partial differential equation representing the problem has a unique solution and that solution depends on the specified initial and boundary conditions.

2 - Question

What is the difficulty in modelling supersonic blunt body problem?

a) The PDE cannot be solved

b) A PDE cannot be formed

c) Mixed flow behaviour

d) Boundary conditions cannot be formed

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Answer: cExplanation: Supersonic flow over a blunt body creates a bow shock. This behaves as a normal shock near the leading edge and as an oblique shock downstream. So, the flow becomes a mixture of subsonic-supersonic flow based on spatial coordinates and makes the mathematical behaviour also mixed.

3 - Question

Which of these mathematical models suit unsteady Navier-Stokes equations?

a) Elliptic

b) Hyperbolic

c) Parabolic

d) Mixed

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Answer: dExplanation: The equations reduced from the Navier-Stokes equations have a particular behaviour. But, the pure unsteady Navier-Stokes equations do not have a particular behaviour. They exhibit mixed behaviour.

4 - Question

Which kind of flows need an initial condition?

a) Parabolic and hyperbolic

b) Hyperbolic and elliptic

c) Elliptic and Parabolic

d) Elliptic

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Answer: aExplanation: Parabolic and hyperbolic equations are marching problems. So, they need initial conditions. Elliptic equations need only boundary conditions.

5 - Question

The flow variables for a/ an _____________ should be solved simultaneously with the flow variables at all other points in the domain.

a) Hyperbolic flow

b) Elliptic flow

c) Parabolic flow

d) Mixed flow

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Answer: bExplanation: Elliptic equations do not march from any condition. They completely rely on the boundary conditions and to solve an elliptic problem, the whole domain should be solved simultaneously.

6 - Question

CFD solutions are excellent for flows with _____________

Note: ‘M’ is the Mach number of the flow.

a) M > 1

b) M < 1

c) M = 1

d) M > 5

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Answer: bExplanation: CFD solutions suit problems with Mach numbers well less than 1. When the Mach number is near or more than 1, they pose difficulties for the solution.

7 - Question

What is the problem in modelling flows with high Reynolds number?

a) A mixture of viscous and inviscid flow regions

b) A mixture of subsonic and supersonic flow regions

c) A mixture of elliptic and parabolic flows

d) A mixture of elliptic and hyperbolic flows

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Answer: aExplanation: At high Reynolds numbers, the viscous regions are very thin. The part of the flow where the boundary conditions are specified behaves different (inviscid) from the part near the body.

8 - Question

Under-specification of boundary conditions gives rise to _____________

a) solution equal to zero

b) no solution

c) a unique solution

d) n-number of solutions

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Answer: dExplanation: If enough boundary conditions are not specified, the number of solutions becomes more. A unique solution to the problem cannot be found

9 - Question

What happens with the over-specification of boundary conditions?

a) Infinite solutions

b) Unique solutions

c) Unphysical solutions

d) No solution

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Answer: cExplanation: When boundary conditions are over-specified, it leads to severe unphysical solutions near the boundaries. It will not reinforce the solution.

10 - Question

What is the problem in solving flows with Mach numbers around and above 1?

a) Mach number

b) Compressibility

c) Continuity

d) Pressure

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Answer: bExplanation: Flows with Mach numbers around and above 1 has the additional issue of compressibility. Air becomes compressible at higher speeds and the effects of compressibility also should be included while solving the problem.