Boundary Layer
aerodynamics, Boundary Layer
Introduction
As long as the angle of attack is small and streamlined flow exists, the potential flow calculations give a pretty accurate estimate of the pressure variation over the surface of the airfoil. This leads to accurate estimation of lift and pitching moment. However, this analysis rests on the assumption that the pressure at the outer edge of the boundary layer is the same as that at the solid surface. This is not entirely self evident. This presumption will be justified by studying the boundary layer. Other obvious aims are to estimate skin friction drag and to give some notions of the boundary layer separation.
Drag caused by viscous effects acts in two ways: skin friction drag and pressure drag. The former is caused by the shear stress at the surface of the body, while the latter is caused by the pressure difference between the front and the rear of the body. The pressure drag is the dominant component of the total drag for streamlined bodies at high angles of attack. The skin friction drag is the dominant component of the total drag for streamlined bodies at low angles of attack. The pressure drag is the dominant component of the total drag for bluff bodies at all angles of attack.
This is also a useful way of defining strealined and bluff bodies.
- Streamlined body
-
A body for which the sking friction drag is the dominant component of the total drag is called a streamlined body.
- Bluff body
-
A body for which the pressure drag is the dominant component of the total drag is called a bluff body.
An airfoil, for instance, is a streamlined body at low angles of attack, since skin friction drag accounts for 80 to 90 % of the total drag. At angles of attack at which the airfoil is stalled, it acts as a bluff body, and the pressure drag accounts for 80 to 90 % of the total drag.
The other components of the drag are induced drag and wave drag. Induced drag is caused by the lift generated by the body. Wave drag is caused by the formation of shock waves in supersonic flow.
Boundary Layer
The boundary layer is a thin layer of fluid adjacent to the surface of a body where the viscous effects are significant. In other words, it is the region next to the wall surface where the flow is significantly retarded due to the influence of the friction between the wall and the fluid. The velocity adjacent to the surface is zero, and it increases to the free stream value at the outer edge of the boundary layer. This defines the velocity boundary layer.
Similarly, we can define the temperature boundary layer (also known as thermal boundary layer), the concentration boundary layer, and the electrical potential boundary layer.
The temperaure boundary layer is the layer of fluid adjacent to the surface of a body where the temperature is different from the free stream value.
The concentration boundary layer is the layer of fluid adjacent to the surface of a body where the concentration of a chemical species is different from the free stream value.
The electrical potential boundary layer is the layer of fluid adjacent to the surface of a body where the electrical potential is different from the free stream value.
In this course, we will only study the velocity boundary layer. Aerospace applications typically involve high Reynolds number flows, and the velocity boundary layer is the most important boundary layer in such flows. Temperature boundary layer is also important in some applications, such as the design of cooling systems for gas turbine engines. This aspect of the boundary layer is studied in the course on heat transfer.
Even though the boundary layer is thin, it has a significant effect on the flow characteristics.
It is interesting to note that the thickness of the velocity boundary layer (\(\delta\)) and the thickness of the temperature boundary layer (\(\delta_T\)) are not the same. In fact, the relative thickness depends on the Prandtl number (\(Pr\))
- Prandtl Number
-
Prandtl number is defined as the ratio of the momentum diffusivity to the thermal diffusivity.
\[ Pr = \frac{\nu}{\alpha} \]
where \(\nu\) is the kinematic viscosity and \(\alpha\) is the thermal diffusivity. For air, the Prandtl number is about 0.7.
The thickness of the velocity boundary layer is called the momentum thickness. The pressure in the boundary layer is not constant, and it increases from the free stream value at the outer edge of the boundary layer to the value at the surface. This defines the displacement boundary layer. The thickness of the displacement boundary layer is called the displacement thickness.
The condition that the fluid velocity is zero at the surface is called the no-slip condition. No direct experimental check of the no-slip condition has been made, but it is a reasonable assumption. In low-density flows such as exists at altitudes above 100,000 m, the mean free path of the molecules is large compared to the size of the body. In such cases, the no-slip condition may not be valid. You may read this short and excellent article for more details.
Few Properties
!. Displacement thickness
\[ \delta^* = \int_0^\infty \left( 1 - \frac{U}{U_e} \right) \, dy \]
where \(U_e\) is the velocity just outside the boudary layer. This definition is modified if the flow is compressible as
$$ ^* = _0^( 1 - ) , dy
Read Anderson (2011) section 17.2 for the two interpretations of the displacement thickness. Also read the concept of effective body from this section.
Typcially \(\delta^* \ll \delta\). Also, \(\delta^* \approx 0.3 \delta\).
- Momentum thickness
\[ \theta = \int_0^\infty \left( \frac{U}{U_e} \right) \left( 1 - \frac{U}{U_e} \right) \, dy \]
Similarly, for a compressible flow, this definition gets modified to
\[ \theta = \int_0^\infty \left( \frac{\rho U}{\rho_e U_e} \right) \left( 1 - \frac{U}{U_e} \right) \, dy \]
- Shape factor
\[ H = \frac{\delta^*}{\theta} \]
- Skin friction coefficient
\[ C_f = \frac{\tau_w}{\frac{1}{2} \rho U_\infty^2} \]
- Local skin friction coefficient
\[ C_{f_x} = \frac{\tau_w}{\frac{1}{2} \rho U_\infty^2} = \frac{1}{U_\infty^2} \int_0^\infty \tau_{wx} \, dy \]
- Average skin friction coefficient
\[ C_f = \frac{1}{U_\infty^2} \int_0^\infty \tau_w \, dy \]
- Local shear stress
\[ \tau_{wx} = \mu \frac{\partial u}{\partial y} \]
- Average shear stress
\[ \tau_w = \frac{1}{\delta} \int_0^\delta \tau_{wx} \, dy \]
Boundary Layer equations
In order to analyse the boundary layer, lets start with the three dimensional Navier-Stokes equations. The continuity equation for an incompressible flow is
\[ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 \]
The two dimensional momentum equations for a steady flow are \[ \begin{aligned} u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} &= - \frac{1}{\rho} \frac{\partial p}{\partial x} + \nu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right) \\ u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} &= - \frac{1}{\rho} \frac{\partial p}{\partial y} + \nu \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} \right) \\ \end{aligned} \]
The boundary conditions for these equations for a boundary layer flow are
\[ \begin{aligned} u &= 0 \quad \text{at} \quad y = 0 \\ u &= U_\infty \quad \text{at} \quad y = \infty \\ v &= 0 \quad \text{at} \quad y = 0 \\ v &= 0 \quad \text{at} \quad y = \infty \\ \end{aligned} \]
The boundary conditions for the pressure are
\[ \begin{aligned} p &= p_s \quad \text{at} \quad y = 0 \\ p &= p_\infty \quad \text{at} \quad y = \infty \\ \end{aligned} \]
The boundary conditions for the velocity components are obtained from the no-slip condition. The boundary conditions for the pressure are obtained from the fact that the pressure at the outer edge of the boundary layer is the same as that at the solid surface.
The boundary layer equations are obtained by non-dimensionalising the Navier-Stokes equations. The non-dimensional variables are
\[ \begin{aligned} \eta &= \frac{y}{\delta} \\ \eta &= \frac{u}{U_\infty} \\ \eta &= \frac{v}{U_\infty} \\ \eta &= \frac{p - p_\infty}{\frac{1}{2} \rho U_\infty^2} \\ \end{aligned} \]
The non-dimensional Navier-Stokes equations are
\[ \begin{aligned} \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} &= 0 \\ u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} &= - \frac{\partial p}{\partial x} + \frac{1}{Re} \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right) \\ u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} &= - \frac{\partial p}{\partial y} + \frac{1}{Re} \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} \right) \\ \end{aligned} \]
The boundary conditions for these equations for a boundary layer flow are
\[ \begin{aligned} u &= 0 \quad \text{at} \quad y = 0 \\ u &= 1 \quad \text{at} \quad y = \infty \\ v &= 0 \quad \text{at} \quad y = 0 \\ v &= 0 \quad \text{at} \quad y = \infty \\ \end{aligned} \]
The boundary conditions for the pressure are
\[ \begin{aligned} p &= 0 \quad \text{at} \quad y = 0 \\ p &= 1 \quad \text{at} \quad y = \infty \\ \end{aligned} \]
The solution of the boundary layer equations is obtained by numerical methods. However, the solution can be obtained analytically for some special cases. The most important of these is the Blasius solution for the boundary layer over a flat plate.
Derivation of Blasius Solution
The Blasius solution is obtained by assuming that the boundary layer is fully developed. This assumption is valid for sufficiently large values of \(x\).
Since the boundary layer is fully developed, the velocity components are
\[ \begin{aligned} u &= f'(\eta) \\ v &= 0 \\ \end{aligned} \]
The boundary layer equations are
\[ \begin{aligned} f'' + f f''' &= 0 \\ \end{aligned} \]
The boundary conditions are
\[ \begin{aligned} f &= 0 \quad \text{at} \quad \eta = 0 \\ f' &= 1 \quad \text{at} \quad \eta = \infty \\ \end{aligned} \]
The Blasius solution is
\[ \begin{aligned} \frac{u}{U_\infty} &= 0.332 \sqrt{\frac{\nu x}{U_\infty y}} \\ \frac{v}{U_\infty} &= 0 \\ \frac{p - p_\infty}{\frac{1}{2} \rho U_\infty^2} &= -0.664 \sqrt{\frac{\nu U_\infty}{x y}} \\ \end{aligned} \]