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Quasi-3D Boundary Layer

Note

These equations are obtained from the 3D boundary layer equations by assuming homogeneous flow in the spanwise direction (\(\partial/\partial z = 0\)). They are the starting point for the Falkner-Skan-Cooke derivation.

Assumptions

Inherited from the 3D boundary layer equations

  • Steady, three-dimensional, laminar flow
  • Cartesian coordinates: \(x\) streamwise, \(z\) crossflow (spanwise), \(y\) wall-normal
  • Calorically perfect gas (constant \(\gamma\), \(c_p\), \(\mathrm{Pr}\))
  • Temperature-dependent viscosity (Sutherland or power law)
  • Thin boundary layer (\(\delta \ll L\))

Additional assumption:

\[\frac{\partial}{\partial z} = 0\]

The flow has three velocity components \((u, v, w)\) but no gradients in the spanwise \(z\)-direction.

Reduction

Starting from the 3D boundary layer equations, applying \(\partial/\partial z = 0\) term by term:

Continuity. The \(\partial(\rho w)/\partial z\) term drops:

\[ \frac{\partial (\rho u)}{\partial x} + \frac{\partial (\rho v)}{\partial y} + \cancel{\frac{\partial (\rho w)}{\partial z}} = 0 \]

x-momentum. The \(w\,\partial u/\partial z\) convective term drops:

\[ \rho \left( u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + \cancel{w \frac{\partial u}{\partial z}} \right) = -\frac{\partial p}{\partial x} + \frac{\partial}{\partial y}\!\left( \mu \frac{\partial u}{\partial y} \right) \]

z-momentum. The \(w\,\partial w/\partial z\) convective term drops. Assuming homogeneous flow in the spanwise direction also implies no spanwise pressure gradient (\(\partial p/\partial z = 0\)), so the pressure driving term drops too:

\[ \rho \left( u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y} + \cancel{w \frac{\partial w}{\partial z}} \right) = \cancel{-\frac{\partial p}{\partial z}} + \frac{\partial}{\partial y}\!\left( \mu \frac{\partial w}{\partial y} \right) \]

y-momentum. Unchanged:

\[\frac{\partial p}{\partial y} = 0\]

Energy. The \(w\,\partial T/\partial z\) convective term and the \(w\,\partial p/\partial z\) pressure work term both drop:

\[ \begin{aligned} \rho c_p \left( u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} + \cancel{w \frac{\partial T}{\partial z}} \right) &= \left( u \frac{\partial p}{\partial x} + \cancel{w \frac{\partial p}{\partial z}} \right) + \frac{\partial}{\partial y}\!\left( k \frac{\partial T}{\partial y} \right) \\ &+ \mu \left[ \left(\frac{\partial u}{\partial y}\right)^{\!2} + \left(\frac{\partial w}{\partial y}\right)^{\!2} \right] \end{aligned} \]

Governing Equations

Continuity

\[ \frac{\partial (\rho u)}{\partial x} + \frac{\partial (\rho v)}{\partial y} = 0 \]

x-momentum

\[ \rho \left( u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} \right) = -\frac{\partial p}{\partial x} + \frac{\partial}{\partial y}\!\left( \mu \frac{\partial u}{\partial y} \right) \]

z-momentum

\[ \rho \left( u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y} \right) = \frac{\partial}{\partial y}\!\left( \mu \frac{\partial w}{\partial y} \right) \]

y-momentum

\[\frac{\partial p}{\partial y} = 0\]

Energy

\[ \rho c_p \left( u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} \right) = u \frac{\partial p}{\partial x} + \frac{\partial}{\partial y}\!\left( k \frac{\partial T}{\partial y} \right) + \mu \left[ \left(\frac{\partial u}{\partial y}\right)^{\!2} + \left(\frac{\partial w}{\partial y}\right)^{\!2} \right] \]

Outer Inviscid Flow

Applying \(\partial/\partial z = 0\) to the 3D outer flow relations:

\[-\frac{\partial p}{\partial x} = \rho_e u_e \frac{\partial u_e}{\partial x}\]

The spanwise pressure gradient \(-\partial p/\partial z\) vanishes, consistent with the assumption of homogeneous flow in the spanwise direction. The edge crossflow velocity \(w_e\) may still be nonzero (it is prescribed as a boundary condition).

Relation to 2D System