Computational Fluid Dynamics Multiple Choice Questions on “Euler Equation”.
1. The general transport equation is (frac{partial(rho Phi)}{partial t}+div(rho Phi vec{u})+div(Gamma grad Phi)+S). For Eulerian equations, which of the variables in the equation becomes zero?
a) Γ
b) ρ
c) Φ
d) (vec{u})
Answer: a
Clarification: Γ is the diffusion coefficient in the general transport equation. Diffusion of any property is not included in Eulerian equations. So, Γ=0.
2. Euler equations govern ____________ flows.
a) Viscous adiabatic flows
b) Inviscid flows
c) Adiabatic and inviscid flows
d) Adiabatic flows
Answer: c
Clarification: Euler equations constitute the governing equations of flow for adiabatic and inviscid flows. Here, the dissipative transport of flow properties is neglected.
3. Which of these is the non-conservative differential form of Eulerian x-momentum equation?
a) (frac{partial(rho u)}{partial t}+nabla.(rho uvec{V})=-frac{partial p}{partial x}+rho f_x)
b) (rhofrac{Du}{Dt}=-frac{partial p}{partial x}+rho f_x)
c) (frac{(rho u)}{partial t}=-frac{partial p}{partial x}+rho f_x)
d) (rho frac{partial u}{partial t}=-frac{partial p}{partial x}+rho f_x)
Answer: b
Clarification: Momentum equation excluding the viscous terms gives the Eulerian momentum equation. This can be given by (rhofrac{Du}{Dt}=-frac{partial p}{partial x}+rho f_x).
4. Eulerian equations are suitable for which of these cases?
a) Compressible flows
b) Incompressible flows
c) Compressible flows at high Mach number
d) Incompressible flows at high Mach number
Answer: d
Clarification: Eulerian equations are best suited for examining incompressible flows at high Mach number. They are used to study flow over the whole aircraft.
5. Euler form of momentum equations does not involve this property.
a) Stress
b) Friction
c) Strain
d) Temperature
Answer: b
Clarification: Euler form of equations is for inviscid flows. For inviscid flows, viscosity is zero. So, there are no friction terms involved.
6. There is no difference between Navier-Stokes and Euler equations with respect to the continuity equation. Why?
a) Convection term plays the diffusion term’s role
b) Diffusion cannot be removed from the continuity equation
c) Its source term balances the difference
d) The continuity equation by itself has no diffusion term
Answer: d
Clarification: Diffusion term, in general, is given by div(ΓgradΦ). For the continuity equation, Φ=1. And grad Φ=0. So, the continuity equation by itself has no diffusion term.
7. Which of these equations represent a Euler equation?
a) (rhofrac{Dv}{Dt}=-nabla p+rho g)
b) (rhofrac{Dv}{Dt}=-nabla p+munabla^2 v+rho g)
c) ∇p=μ∇2v+ρg
d) 0=μ∇2v+ρg
Answer: a
Clarification: (frac{rho Dv}{Dt}=-nabla p+rho g) represents a Euler equation. All other equations have this term μ∇2v representing diffusion.
8. Which of the variables in the equation (rhofrac{Du}{Dt}=-frac{partial p}{partial x}+frac{partial tau_{xx}}{partial x}+frac{partial tau_{yx}}{partial y}+frac{partial tau_{zx}}{partial z}+rho f_x) will become zero for formulating Euler equation?
a) fx, τyx, τzx
b) τxx, τyx, u
c) τxx, τyx, τzx
d) τxx, p, τzx
Answer: c
Clarification: τxx, τyx, τzx represent shear stresses due to viscous effects; u is the x-velocity; fx is the body force and p is the pressure. τxx,τyx,τzx should become zero for the flow to be in-viscid and the equations to be Eulerian.
9. In Euler form of energy equations, which of these terms is not present?
a) Rate of change of energy
b) Heat radiation
c) Heat source
d) Thermal conductivity
Answer: d
Clarification: As the flow considered by Euler equations is adiabatic, heat cannot enter or exit the system. So, the thermal conduction is omitted.
10. To which of these flows, the Euler equation is applicable?
a) Couette flow
b) Potential flow
c) Stokes Flow
d) Poiseuille’s flow
Answer: b
Clarification: Among the given flows, only potential flows are in-viscid. So, the Euler equation is applicable to only potential flows among the above.