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).