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Introduction to CFD (ME 423): Exam 2

I. PROBLEM 1

1,1

0,0

Figure 1: A 2D square channel.

Consider a square two-dimensional channel through which a ﬂuid is ﬂowing at uniform velocity U in the x-direction (see Fig. 2). The channel dimensions are x = [0, 1] , y = [0, 1]. The ﬂuid enters the channel at x = 0 with a uniform temperature of 0 and leaves with a temperature Texit(y) at x = 1. The bottom and top walls satisfy Neumann boundary conditions, oy=0= -3x2 and

oy=1= x2 + 4x, respectively [remember that direction matters for normal boundary conditions so be careful with signs]. There is a heat source S(x, y). The steady state governing equation for temperature is given by

U = + S(x, y).

(1)

Taking

U = 1

and

S(x, y) = y2 + 5xy - 4x,

solve for the temperature ﬁeld in the domain and answer the following questions:

1. (10 pts .) Discuss under what assumptions would you get Eq. 1 above from the general convection-diﬀusion equation for temperature.

2. (40 pts .) Use a central diﬀerence scheme to obtain a numerical solution for T from Eq. 1. Show your derivation of the discretized form of the equation. Write a MATLAB code to solve for T and show a contour plot of temperature distribution inside the domain.

3. (15 pts .) Plot the temperature proﬁle Texit(y) at the exit of the channel. Fit a curve to this proﬁle and provide the expression. You can use MATLAB’s curve-ﬁtting toolbox (launched by typing “cftool” in the command window) to ﬁt an nth degree polynomial for this purpose. This will familiarize you with curve ﬁtting in MATLAB.

4. (15 pts .) Plot the temperature proﬁle Ty=0.5 (x) at the middle of the channel. Fit a curve to this proﬁle and provide the expression.

5. (15 pts .) Does the temperature proﬁle become fully developed in this problem? We will consider a fully developed proﬁle to be one for which = 0 beyond a certain length of the channel. Justify your answer.

II. PROBLEM 2

You have been taught about many numerical schemes in class. In this problem, we will introduce the concept of numerical diﬀusion.

Consider the following PDE

+ V = 0,

where V is constant.

1. (10 pts .) Step 1: Verify that the above equation a hyperbolic solution of the form u = g(x - Vt) is a solution to this equation. Physically, what does this solution show? [Hint: Think how the solution changes with time. Simply say what happens to the shape of the solution.]

2. (5 pts .) Write the Taylor series expansion for u(x + ∆x, t) around u(x, t) up to and including the (∆x)2 term. This expansion should be done with respect to x. Keep the and terms intact i.e. don’t replace them with any approximations.

3. (5 pts .) Rearrange the above expression to get an approximation for

4. (5 pts .) Substitute this expression for into the original PDE. Clearly write this new version of the PDE. [Some extra info – The form of the PDE that you get after replacing

with the Taylor Series expansion is called the MODIFIED DIFFERENTIAL EQUATION. You are encouraged to read up on this topic and understand what it says about numerical approximations of a PDE.]

5. (10 pts .) Explain how this version of the PDE is diﬀerent from the original and where the diﬀusion is coming from and what the diﬀusion coeﬃcient is. What happens to the numerical diﬀusion when you decrease the mesh size? What happens to the numerical diﬀusion if V is higher?

6. (20 pts .)

Figure 2: Grid point cluster for the one-dimensional domain.

Consider an initial condition for the solution of Eq. 2 as the Heaviside function deﬁned as follows:

u(x, t = 0) = H(x) =．．0,

．(1,

x < 0

x ≥ 0

Plot the function above and notice that it has a sharp change at x = 0. Choose V = 1 and

∆t/∆x = 0.5. Evolve Eq. 2 using an explicit scheme with backward diﬀerence in x as shown

below:

uP(n)+1 = uP(n) - V (uP(n) - uW(n)) ,

where where the subscripts correspond to the locations shown in Fig. 2. The superscripts refer to time instants (n and n+1 refer to the previous and current time instants respectively). Plot the solution at multiple time instants in the same plot and show that the initial sharp change begins to smooth out as the solution travels forward with time.

III. PROBLEM 3

In this problem, we will use von Neumann stability analysis on the explicit and implicit schemes of solving Eq. 2 given in the previous problem. In class, you have been taught to approximate the round-oﬀ error as a summation of sine modes, i.e. ε(x, t) = m(北)in Cjeajtsin(kjx) (see class notes). This is because the problem discussed in class had 2nd order derivatives in the governing equation, and T = 0 boundary conditions on a ﬁnite domain. In this problem, we will consider an inﬁnite domain, and for that, the correct Fourier expansion needs to include both sine and cosine modes. Therefore, we need to approximate the round-oﬀ errors as follows:

ma北

ε(x, t) = Cjea t ik xjej ,

j =λmin

where i = ^-1, λmin = 2∆x and λmax = >.

1. (15 pts .) Consider a simple explicit scheme for solving Eq. 2 using central diﬀerencing in

space as shown below:

uP(n)+1 = uP(n) - V ￥、 ,

where the subscripts correspond to the locations shown in Fig. 2. The superscripts refer to time instants (n and n + 1 refer to the previous and current time instants respectively). The speed of the wave V is constant. Show that this scheme is unstable.

2. (15 pts .) Consider an implicit scheme for solving Eq. 2 as follows: uP(n)+1 = uP(n) - V ╱ \ .