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...

solution with full numerical steps


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) Figure 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 H H U U U 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 turbulence 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 turbulent 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 turbulence model commonly used in CFD? Describe the limitations of this model and present alternative two-equation turbulence models. [10 Marks] (iv) In turbulent flow, how is the boundary layer resolved in the near-wall region? Discuss the possible approaches in linking the wall effect with the bulk turbulent flow. [10 Marks] END OF EXAM