QUESTION 1 [50 Marks] PART A (25 Marks) Consider a parallel plate flow problem as illustrated in Fig.1(a) where the normal and tangential stresses acting on an infinitesimal control volume as shown...

QUESTION 1 [50 Marks] PART A (25 Marks)
Consider a parallel plate flow problem as illustrated in Fig.1(a) where the normal and tangential stresses acting on an infinitesimal control volume as shown along the Cartesian direction of x are shown in Fig.1(b):
Figure 1(a) XXXXXXXXXXFigure 1(b) The mass conservation equation is given as:
(
1
)
 (u)
(v)
(w)
+ + + =0
(1)
t x y z
(i)    Provided that the flow is incompressible, reduce the above equation. What physical description of the flow can be concluded based on the reduced equation?
[6 Marks] (ii) Based on the finding in (i) and given the fluid is Newtonian with constant fluid properties,
and no external body forces, derive the following x-momentum equation and explain
the physical meaning of each terms:
 u
t
 u
+u x
+v u
y
u 
+w z 
=−p +
x
2u
x2
2u
+ +
y2
2u 
z2 
(2)
   
The normal and tangential viscous stresses are given as:
 =−p +2u +u + v + w; 
=u +v 
u
w
xx x
x y
z  yx
y
x  and zx =z + x 
     
[9 Marks]
(iii) Starting from the following three-dimensional x-momentum equation:
u u
u p
2u
2u
2u 
u +v
+w
=− + + + 
(3)
x y
z x
x2
y2
z2 
Non-dimensionalize the above equation using the following scaling factors:
x* = x ;
y* = y
; z* = z ; u* = u ; v* = v
; w* = w
and
p* = p     
H XXXXXXXXXXH XXXXXXXXXXH XXXXXXXXXXU XXXXXXXXXXU XXXXXXXXXXU
U 2
[5 Marks]
(iv) Provided that there are two separate flows across two parallel plates with different fluid viscosities ? but sharing the same Reynolds number. Briefly discuss about the outlet velocity profiles. You may use figure illustration to justify your answer.
[5 Marks]
PART B (25 Marks)
The one-dimensional temperature transport equation for pure convection is given as:
T +u T =0
t x
(4)
(i)    Based on the above equation, write down the Finite Difference Approximation (FDA) with explicit formulations, using forward difference in time and backward upwind difference (i.e. u > 0) in space for the convection term.
[6 Marks] (ii) Apply the Von Neumann stability analysis to FDA written in part (i).
[14 Marks]
(iii) Based on your analysis in part (ii), justify whether this numerical scheme is unstable, stable or unconditionally stable? If stable, please specify the stability criterion.
[5 Marks]
QUESTION 2 [50 Marks]
(i)    Describe the major difference between DNS, LES and RANS methods for tu
ulence closure.
[6 Marks]
(ii) In practical CFD calculations, a system of time-averaged equations can be obtained using either the Reynolds-Averaged Navier-Stokes (RANS) approach or Favre Averaged Navier Stokes (FANS) to modelling tu
ulent flow
 ( u )  ( uu )  ( uv )
+ + =
 u +
 u −p
(5)
t x
y x 
x 
y 
y  x
   
Applying the Reynolds averaging rule such as  =
;  + = +
;  = ;
 = +; and
 = 
s s
, derive the Reynolds-Averaged Navier-Stokes (RANS)
form of the above Equation:
[10 Marks]
(
,
)If a mean property can be defined as ?̃ = ?? derive the Favred-Averaged Navier-Stokes
?
(FANS) form of the above Equation:
[10 Marks]
Comment on the difference between the RANS and FANS equations for negligible density fluctuations.
[4 Marks]
(iii) What is a typical two-equation tu
ulence model commonly used in CFD? Describe the limitations of this model and present alternative two-equation tu
ulence models.
[10 Marks] (iv) In tu
ulent flow, how is the boundary layer resolved in the near-wall region? Discuss the
possible approaches in linking the wall effect with the bulk tu
ulent flow.
[10 Marks]
END OF EXAM