ME 431A/538A/538B Homework 22 October 2018 Advanced Fluid Mechanics For Friday, October 26 th Start reading the handout entitled Notes on finite-volume methods. Review Chapter 7 on Dimensional Analysis No homework assignment on Monday October 28 th, as the mid-term is held Friday November 2 nd Due Monday, October 29 th Problem 1. Consider the problem of steady, two-dimensional, incompressible, laminar flow past a heated cylinder, as shown in Figure 1. It is assumed that the temperature field T (x, y) satisfies the heat equation: T t + V T = k 2 T, (1) plus the following boundary conditions on T : T (x, y) = T i at the inlet, T (x, y) = T c on the surface of the cylinder, and T y = 0 along the top and bottom surfaces. It is assumed that the thermal diffusivity k is constant, and that the temperature fluctuations are not large enough to affect the flow field. T i T c T/ y = 0 T/ y = 0 Figure 1: Flow past a heated cylinder. 1. If T s (x, y) is a solution to this problem, then show that T (x, y) = T s (x, y) T i is a solution to the same problem, but with inlet conditions T (x, y) = 0 and T (x, y) = T c T i on the surface of the cylinder. 2. Show that T (x, y) = T s(x, y) T i is a solution to the same problem, but with T (x, y) = 0 at T c T i the inlet and T (x, y) = 1 at the surface of the cylinder. 1
3. Argue that if one solves Equation (1) subject to the inlet condition T (x, y) = 0, the condition of T (x, y) = 1 on the heated surface, and the condition T = 0 along the top and bottom y boundaries, then this solution can be used for problems with any inlet values of T i, any values of T c on the cylinder surface, and no heat flux at the top and bottom boundaries. This property is due to the fact that Equation (1) for T is linear in T. Problem 2. Consider the problem of two-dimensional, steady, incompressible, fully-developed flow in a channel as discussed in class and in the text. Assume that the flow entering the channel has a temperature distribution of T (0, y) = F (y), h/2 y h/2, where F (y) is an arbitrary function of y. Are there either constant temperature or constant temperature flux boundary conditions along the channel side walls such that the temperature field would also be fully-developed, i.e., independent of x? Problem 3. Computer Simulation of Steady Flow past a Cylinder. Consider two-dimensional, steady, laminar flow past a circular cylinder. The objectives of this problem are to use STAR-CCM+ to address the case of flow past a circular cylinder for a range of Reynolds numbers, and in particular to determine the drag force and the extent of the recirculation zone for each case computed. y = H u = U v = 0 P i (y) slip v=0, u/ y=0 V = 0 u(y) v(y) P 0 d slip y = H x = L x = 0 x = L x = 2L Figure 2: Sketch of the computational domain. The physical/computational domain is shown in Figure 2. A circular cylinder of diameter d is placed with its axis perpendicular to the flow direction in a stream with speed U. The flow is assumed to be laminar, steady, and two-dimensional. The computational domain ranges from L x 2L, and H y H. for graduate students 2
The boundary conditions are listed on the sketch. The boundary condition at the inlet is uniform (constant) velocity, i.e., V = (u, v) = (U, 0), where U is the constant inlet velocity. The boundary conditions on the surface of the cylinder are no-slip, i.e., V = (0, 0). In order to eliminate boundary layers on the top and bottom walls, take the boundary condition to be free-slip (no shear stress), i.e., ( u v = 0 y + v ) = u x y = 0 ; the latter is because v = constant = 0 along the boundary. Finally take the boundary condition at the outlet to be constant pressure. For this computational problem take the fluid to be water, and use the default values for the density and viscosity used in STAR-CCM+, i.e., ρ = 997.561 kg/m 3, and = 8.8871 10 4 Pa s. Take the cylinder radius to be 1 cm, and the size of the computational domain given by L = 10 cm and H = 10 cm. Perform simulations for the following cases: Re = ρud = 13.1 and 41.0. In performing the simulations, it is suggested (although not mandatory) that you use the computational domain shown. For each Reynolds number case, provide the following information from your simulations: 1. a streamline plot of the overall flow field, emphasizing the flow nearer the cylinder if necessary. 2. a plot of u versus y at the outflow to the computational domain. 3. a plot of u versus x along the centerline y = 0 behind the cylinder, which will determine the length of the recirculation zone. 4. a comparison of the ratio of the length of the recirculation zone to the cylinder diameter, obtained from your calculations, with that obtained by flow visualization given in the figures below. 5. from the simulation, determine the drag force on the cylinder. Finally, for the calculation of the drag force D, form the nondimensional drag (the drag coefficient) and plot these as points on a copy of Figure 9.14, page 186, of the text. Figures 3 and 4 below, taken from the book by van Dyke (An Album of Fluid Motion, Parabolic Press), can be used to estimate the ratio of the length of the recirculation zone to the cylinder diameter. Note that for this laminar, incompressible flow, the length of the recalculation zone l should depend on the fluid density, ρ, the fluid viscosity,, and oncoming flow speed, U, and the cylinder diameter, d, i.e., l = f(ρ,, U, D). 3
In this problem there are five dimensional quantities, l, ρ,, U, and d, and three dimensions, M (mass), L (length), and T (time). So from the Buckingham Pi theorem, there should be two non-dimensional parameters. One choice of these is: Π 1 = l d and Π 2 = ρud, the Reynolds number. Buckingham s theorem states that Π 1 should be a function of Π 2, i.e., Π 1 = f(π 2 ), but it does not give the function form for g. Therefore we know that ( ) l ρud d = g. Therefore, for two different flows at the same Reynolds number ρud/, then l/d should also be the same. This is used to determine l/d, and hence l, for each case. Figure 3: Photograph of circular cylinder flow at Re = 13.1. The cylinder is moving through a tank of water containing aluminum powder, an dis illuminated by a sheet of light from below the free surface. 4
Figure 4: Photograph of circular cylinder flow at Re = 41.0. 5