Boundary Layer Equations

The boundary layer equations represent a significant simplification over the full Navier-Stokes equations in a boundary layer region. The simplification is done by an order-of-magnitude analysis; that is, determining which terms in the equations are very small relative to the other terms.

For simplicity, we will determine the boundary layer equations for steady, incompressible ($M \ll 1$), uniform flow over a flat plate. The equations for flow over curved surfaces, and for nonuniform flow, differ only slightly, in that they need to account for changes in $U$along the wall surface. Curved surfaces also require a transformation to surface coordinates. The equations for compressible boundary layers are somewhat more complex than the incompressible equations; we will not consider them here.

Assumptions

The boundary layer equations require several assumptions about the flow in the boundary layer.

1.

  All of the viscous effects of the flowfield are confined to the boundary layer, adjacent to the wall. Outside of the boundary layer, viscous effects are not important, so that flow can be determined by inviscid solutions such as potential flow or Euler equations.

2.

  The viscous layer is thin compared to the length of the wall. If $L$ is a characteristic length of the the wall, then $\delta/L \ll 1$. Also, $x=O(L)$ and $y=O(\delta)$. This assumption is obviously not valid near the leading edge of the wall; other methods (such as stagnation flow) are used to determine the upstream boundary condition.

3.

  The boundary conditions of the boundary layer region are the no-slip condition at the wall, and the free-stream condition at infinity:

$$u(x,0)=0\qquad v(x,0)=0$$

$$u(x,\infty)=U\qquad v(x,\infty)=0$$

4.

  In the boundary layer, $u=O(U)$.

Order-of-Magnitude Analysis

For this problem, we consider a two-dimensional, semi-infinite flat plate with its leading edge at the origin. The plate coincides with the positive $x$-axis. The inviscid flow field is steady, uniform, parallel to the flat plate, with a velocity of $U_\infty$, a density of $\rho_\infty$, and a dynamic viscosity of $\mu_\infty$.

The continuity and momentum conservation equations for incompressible, two-dimensional flow are:

$$\frac{\partial u}{\partial x}+ \frac{\partial v}{\partial y} = 0$$ (3)

$$\rho\left(u\frac{\partial u}{\partial x} +v\frac{\partial u}{... ...l^2 u}{\partial x^2} +\frac{\partial^2 u}{\partial y^2}\right)$$ (4)

$$\rho\left(u\frac{\partial v}{\partial x} +v\frac{\partial v}{... ...l^2 v}{\partial x^2} +\frac{\partial^2 v}{\partial y^2}\right)$$ (5)

To make the order-of-magnitude analysis more straightforward, we shall nondimensionalize the equations. Define nondimensional variables, denoted by an asterisk (${u^*}$, ${v^*}$, etc.), by the following:

$${u^*} = \frac u{U_\infty} \qquad {v^*} = \frac v{U_\infty} \qquad {x^*} = \frac xL \qquad {y^*} = \frac yL$$

$${\rho^*} = \frac \rho{\rho_\infty} \qquad {p^*} = \frac p{\rh... ..._\infty} \qquad Re = \frac{\rho_\infty U_\infty L}{\mu_\infty}$$

These definitions, substituted into Equations 3-4, yield the nondimensional continuity and momentum equations:

$$\frac{\partial{u^*}}{\partial{x^*}}+ \frac{\partial{v^*}}{\partial{y^*}} = 0$$ (6)

$${u^*}\frac{\partial{u^*}}{\partial{x^*}} +{v^*}\frac{\partial... ...ial {x^*}^2} +\frac{\partial^2 {u^*}}{\partial {y^*}^2}\right)$$ (7)

$${u^*}\frac{\partial{v^*}}{\partial{x^*}} +{v^*}\frac{\partial... ...ial {x^*}^2} +\frac{\partial^2 {v^*}}{\partial {y^*}^2}\right)$$ (8)

In addition, we define a nondimensional boundary layer thickness ${\delta^*}=\delta/L$ (not to be confused with the displacement thickness; as this analysis does not use the displacement thickness). By Assumption 2 above, ${\delta^*} \ll 1$.

 

Based on Assumption 4, ${u^*}=O(u/U)=O(1)$. And based on Assumption 2, ${x^*}=O(x/L)=O(1)$ and ${y^*}=O(y/L)=O({\delta^*})$. From the continuity equation, Equation 6,

$$\frac{O(1)}{O(1)}+\frac{{v^*}}{O({\delta^*})} = 0$$

which means that ${v^*}=O({\delta^*})$. This in turn means that the $y$-component of velocity is small in the boundary layer.

From the $x$-momentum equation, Equation 8,

$$O(1)\frac{O(1)}{O(1)} +O({\delta^*})\frac{O(1)}{O({\delta^*})... ...Re}\left(\frac{O(1)}{O(1)}+\frac{O(1)}{O({\delta^*}^2)}\right)$$

Note that we have made no assumptions about pressure, and so we do yet not know its order of magnitude. For now, we keep the pressure gradient term.

Considering the two viscous terms, we see that the first viscous term, which is of order $1/Re$, is very small compared to the second term, which is of order $1/(Re\,{\delta^*}^2)$. We therefore drop the first term, which corresponds to the term $\partial{u^*}/\partial{x^*}$ in Equation 7.

According to Assumption 1, the region near the wall is a viscous boundary layer. In order to keep the remaining viscous term important (but not have it dominate), the term must be of order one, to match the order of the convective terms. Thus, $Re$ must be of order $1/{\delta^*}^2$.

We see, then, that the assumption of a thin boundary layer is reasonable only for flows where $Re \gg 1$. Fortunately, this is true of many flows. However, for flows of very viscous or very slow fluids, the thin boundary layer is not a reasonable assumption.

Now, we expect any changes in pressure along the boundary layer to affect the boundary layer. Therefore, we assign pressure an order of magnitude of 1 so that $\partial{p^*}/\partial{x^*} = O(1)$ to match the other terms. (There will not be any pressure changes for the flat plate, but we keep this term in the interests of generality.)

Now, from the $y$-momentum equation, Equation 8,

$$O(1)\frac{O({\delta^*})}{O(1)} +O({\delta^*})\frac{O({\delta^... ...delta^*})}{O(1)} +\frac{O({\delta^*})}{O({\delta^*}^2)}\right)$$

The pressure gradient term is of order $1/{\delta^*}$. Every other term in this equation is at most of order $\delta^*$. All of the other terms are very small compared the to the pressure gradient term, and to the terms from the continuity and $x$-momentum equations. Hence, the pressure gradient term is the only term that is retained from Equation 8.

Equation 8 reduces to $\partial{p^*}/\partial{y^*}=0$. This says that the pressure difference across the boundary layer is essentially zero. This is why inviscid solutions have reasonable success: they can predict the pressure distribution on a surface with considerable accuracy because the boundary layer does not affect the pressure that much.

So, after the order of magnitude analysis, Equations 6-8 become:

$$\frac{\partial{u^*}}{\partial{x^*}}+ \frac{\partial{v^*}}{\partial{y^*}} = 0$$ (9)

$${u^*}\frac{\partial{u^*}}{\partial{x^*}} +{v^*}\frac{\partial... ...l{x^*}} +\frac 1{Re} \frac{\partial^2 {u^*}}{\partial {y^*}^2}$$ (10)

$$0 = \frac{\partial{p^*}}{\partial{y^*}}$$ (11)

The continuity equation does not simplify at all. The $x$-momentum equation, however, loses a second order term. In the process, it changes from elliptic to parabolic in nature. Because parabolic equations are easier to solve numerically than elliptic equations, this is very significant simplification. Also, the $y$-momentum equation basically disappears, which is a tremendous simplification.

The boundary layer equations can be used in dimensional form as well:

$$\frac{\partial u}{\partial x}+ \frac{\partial v}{\partial y} = 0$$

$$\rho\left(u\frac{\partial u}{\partial x} +v\frac{\partial u}{... ...{\partial p}{\partial x} +\mu\frac{\partial^2 u}{\partial y^2}$$

$$0 = \frac{\partial p}{\partial y}$$

Although these equations have been obtained by considering uniform flow over at a flat plate, they are valid for curved surfaces also, as long as the curvature of the surface is not too great. Instead of using Cartesian coordinates $x$ and $y$, the surface coordinates $s$ and $n$are used on curved surfaces. Also, the pressure gradient term, $\partial p/\partial x$, is not zero for curved surfaces.