UCONN ME 3255 Computational Mechanics Page 1 of 3 Final Project 12 questions, 28 points A major design parameter affecting thermal efficiency of gas turbines is the temperature of the hot gases resulting from the combustion process. Making substantial increase in thermal efficiency relies on increasing the combustion product temperature. However, this temper- ature is limited by the endurance of turbine components, including the blades. Developing advanced cooling systems is therefore a critical issue that needs to be addressed in the design phase to ensure that the turbine blades can endure high temperature levels. The trailing edge region of the blade is of particular importance as it needs to be kept thin to limit the aerodynamic losses, making it the most vulnerable part of the blade at high temperatures. The state-of-the-art cooling systems for the trailing edge is based on film cooling, consisting of flow of cooler air, coming from blade internal cooling, through slots at the trailing edge, as shown in Fig. 1, to cool down this region. Figure 1: Film cooling of the trailing edge of a gas turbine blade. The objective of this project is to analyze some of the aerodynamic and heat transfer charac- teristics of a laminar wall jet which is an idealized flow configuration relevant to film cooling of trailing edge region of gas turbine blades as well as many other application in, e.g., aerofoil design, combustion chamber wall cooling and air conditioning. Wall jet consists of a jet of fluid emanating from a slot near a solid wall (Fig. 2) and spreading next to the wall as a shear layer. I Aerodynamics Consider a laminar wall jet of air with uniform inlet velocity of Ui = 1m/s, the kinematic viscosity is ν = 1.5 × 10−5m2/s and the density is ρ = 1.2 kg/m3. The flow has a Reynolds number of Rex = Uix ν which is considered to be smaller than the critical limit and hence, the flow is in the laminar regime; u and v denote the velocity components in x and y directions, respectively. Such flow is governed by partial differential equations (PDEs) corresponding to incompressible form of Navier-Stokes and the continuity equations. Introducing non- dimensional y coordinate, η = y x (Rex) 1/4, and velocity f ′(η) = df dη = u Uo , where Uo Ui = 4 Re 1/2 x , these PDEs are transformed into the following ordinary differential equation (ODE) which is much easier to solve, f ′′′ + ff ′′ + 2f ′2 = 0 (1) UCONN ME 3255 Computational Mechanics Page 2 of 3 Figure 2: Schematic of a laminar wall jet. subject to boundary conditions η = 0 : f = f ′ = 0 η = H : f ′ = f ′′ = 0 (2) where H refers to far field in y direction (y →∞). Here, we set H = 10. (a) (2.5 pts) An analytical solution to this ODE is obtained in implicit form as η = ln √ 1 + √ f + f 1−√f + √ 3 arctan √ 3f 2 + √ f (3) Use this expression to find f(η) (note that 0 ≤ f < 1). Plot f(η) for η in the range [0, H ]. (b) (2 pts) Find f ′(η) with at least second order accuracy and plot it vs. η. (c) (2 pts) Find f ′′(η) with at least second order accuracy and plot it vs. η. Show that the skin friction coefficient Cf = τw 1 2 ρU2 ∞ ≈ 1.778 Re 5/4 x as obtained from theory. τw is the shear stress at the wall τw = µ ∂u ∂y y=0 = ρU 2 ∞ Re 5/4 x 4f ′′(0). (d) (2 pts) Find maximum velocity f ′max = umax/Uo value and its η location. Verify that f ′max ≈ 2−5/3 which occurs at η ≈ 2.029 consistent with the theoretical values. (e) (2 pts) Calculate the wall jet momentum flux and show that it is consistent with the theoretical value:(∫ ∞ 0 u2dy ) (∫ ∞ 0 udy ) = 43Uiν 2 (∫ H 0 f ′2dη )(∫ H 0 f ′dη ) ≈ 128 9 Uiν 2. (f) (2 pts) Find the shear layer thickness δ1 defined as the η location where f ′(η) ≈ 0.01. Show that your prediction is in agreement with theoretical value δ1 ≈ 6.72 (g) (0.5 pts) Calculate the v velocity at the edge of the layer, vH = (3ηf ′−f)Ui(Rex)−3/4 at η = H and verify that your result matches the theoretical value vH ≈ (−1)Ui(Rex)−3/4. The negative vH value indicates that the ambient fluid gets sucked into the shear layer as the jet develops downstream (called entrainment). UCONN ME 3255 Computational Mechanics Page 3 of 3 (h) (4 pts) Solve Eq. (1) for f(η) and compare your solution with that obtained in Part (a). Implement the boundary condition at η = H as f ′ + f ′′ = 0. II Heat transfer The shear layer formed by the wall jet causes variation of temperature near the wall resulting in film cooling of the wall (Fig. 3). The convective heat transfer from the wall is governed by a PDE for temperature corresponding to energy equation. Using the non-dimensional variables above along with Θ(η) ≡ T−T∞ Tw−T∞ we transform this PDE into an ODE for Θ d2Θ dη2 + Pr f dΘ dη = 0 (4) subject to boundary conditions η = 0 : Θ = 1 η = H : Θ = 0 (5) where Pr is the Prandtl number, Tw is the temperature of the surface and T∞ is the inlet and ambient air temperatures. We set Pr = 0.7. Figure 3: Thermal boundary layer over a flat plate. (i) (4 pts) Solve this ODE to find Θ(η) as a function of η in the range [0, H ]. Plot Θ(η). (j) (2 pts) Calculate the thickness of the thermal boundary layer δT (i.e., η location where Θ ≈ 0.01) and show that δT ≈ δ1(Pr)−1/3 as expected from theory. (k) (0.5 pts) Calculate the temperature gradient at the wall Θ′(0) = dΘ dη η=0 with at least second order accuracy and show that the predicted Nusselt number compares reasonably well with the theoretical value Nu Re 1/4 x = −Θ′(0) ≈ 0.235(Pr)1/3. (l) (4.5 pts) Solve Eq. (4) with its boundary conditions using finite-difference method. Show verification of your finite-difference solution by demonstrating that the solution becomes less sensitive to step size ∆η as the step size value decreases (due to decrease in truncation error); this can be done by plotting Θ versus η for at least three different ∆η values. Compare your solution with that obtained in Part (i).