Computational Fluid Dynamics Multiple Choice Questions on “Governing Equations – Reynolds Transport Theorem”.
1. The Reynolds transport theorem establishes a relationship between __________ and ___________
a) Control mass system, Control volume system
b) Differential equation, Integral equation
c) Non-conservative equation, Conservative equation
d) Substantial derivative, Local derivative
Answer: a
Clarification: Equations formed by considering the control mass system and control volume system are not the same even if the same physical law is used. A relation between these equations is established by Reynolds transport theorem.
2. Let B denote any property of a fluid flow. The statement of Reynolds transport theorem is “The instantaneous total change of B inside the _____________ is equal to the instantaneous total change of B within the ______________ plus the net flow of B into and out of the _____________”
a) Control volume, Control mass, Control volume
b) Control volume, Control volume, Control mass
c) Control mass, Control mass, Control volume
d) Control mass, Control volume, Control volume
Answer: d
Clarification: Statement of Reynolds Transport Theorem: “The instantaneous total change of B inside the control mass is equal to the instantaneous total change of B within the control volume plus the net flow of B into and out of the control volume”.
3. Consider the following terms:
MV → Material Volume (Control Mass)
V → Control Volume
S → Control Surface
B → Flow property
b → Intensive value of B in any small element of the fluid
ρ → Density of the flow
t → Instantaneous time
( vec{v} ) → Velocity of fluid entering or leaving the control volume
( vec{n} ) → Outward normal vector to control surface
Which of these equations is the mathematical representation of Reynolds transport theorem in the above terms?
a) ((frac{dB}{dt})_{MV} = frac{d}{dt}(int_sb rho dS) + int_vb rho vec{v}.vec{n} dV)
b) ((frac{dB}{dt})_{MV} = frac{d}{dt}(int_vb rho dV) + int_sb rho vec{v}.vec{n} dS)
c) ((frac{dB}{dt})_V = frac{d}{dt}(int_{MV}b rho MV) + int_sb rho vec{v}.vec{n} dS)
d) ((frac{dB}{dt})_{MV} = int_vb rho dV + frac{d}{dt}(int_sb rho vec{v}.vec{n} dS))
Answer: b
Clarification:
((frac{dB}{dt})_{MV} →) Instantaneous total change of B in material volume
(frac{d}{dt} (int_vb rho dV) → ) Instantaneous total change of ” B” within control volume
( int_sb rho vec{v}. vec{n}dS → ) Net flow of B into and out of the control volume through control surfaces
Reynolds transport theorem states:
(Instantaneous total change of B in material volume)=(Instantaneous total change of B within control volume + Net flow of B into and out of control volume through control surfaces)
Therefore,
((frac{dB}{dt})_{MV} = frac{d}{dt}(int_vb rho dV) + int_sb rho vec{v}.vec{n}dS).
4. Leibniz rule is applied to which of these terms in deriving Reynolds transport theorem?
a) Volume integral term of control volume
b) Differential term of material volume
c) Surface integral term of control volume
d) Volume integral term of material volume
Answer: a
Clarification: Using Leibniz rule, the differentiation of an integral term can be reduced. Here, differential of integral exists in the Volume integral term of Control Volume which is given by (frac{d}{dt}(int_vb rho dV).)
5. Why a surface integral is used to represent flow of B into and out of the control volume?
a) Control volume is moving
b) Flow of fluid is through the control surfaces
c) Fluid only on the control surfaces
d) Control volume is stationary
Answer: b
Clarification: Fluid can enter into or exit from the control volume through the control surface. If this flow velocity is integrated along the control surfaces, we can get the net inflow or outflow of fluid to the control volume.
6. When is Leibniz rule applicable to control volume?
a) When control volume is moving
b) When control volume is deforming
c) When control volume is fixed
d) In all conditions