2d equations
Assumptions
- Steady, two-dimensional, laminar flow
- Calorically perfect gas (constant \(\gamma\), \(c_p\), \(\mathrm{Pr}\))
- Temperature-dependent viscosity (Sutherland or power law)
- Thin boundary layer (\(\delta \ll L\))
Governing Equations
The compressible laminar boundary layer equations for a calorically perfect gas are (see for example 12):
Continuity
\[
\frac{\partial (\rho u)}{\partial x} + \frac{\partial (\rho v)}{\partial y} = 0
\]
x-momentum
\[
\rho \left( u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} \right)
= -\frac{dp}{dx} + \frac{\partial}{\partial y}\!\left( \mu \frac{\partial u}{\partial y} \right)
\]
y-momentum
\[
\frac{\partial p}{\partial y} = 0
\]
Energy
\[
\rho c_p \left( u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} \right)
= u \frac{dp}{dx}
+ \frac{\partial}{\partial y}\!\left( k \frac{\partial T}{\partial y} \right)
+ \mu \left(\frac{\partial u}{\partial y}\right)^{\!2}
\]
Perfect gas state equation
\[
p = \rho R T
\]
From the y-momentum equation it follows that pressure is uniform across the layer and imposed entirely by the outer inviscid flow (see derivation below).
Outer Inviscid Flow
The pressure gradient is set by the inviscid outer flow via the Euler momentum equation:
\[
-\frac{dp}{dx} = \rho_e u_e \frac{du_e}{dx}
\]
Derivation from the Navier-Stokes equations
Compressible Navier-Stokes Equations (2D)
To obtain the boundary layer equations, start with the compressible Navier-Stokes (NS) equations for a Newtonian fluid with Stokes' hypothesis (\(\lambda = -2\mu/3\)).
Continuity
\[
\frac{\partial \rho}{\partial t}
+ \frac{\partial (\rho u)}{\partial x}
+ \frac{\partial (\rho v)}{\partial y} = 0
\]
x-momentum
\[
\rho \left(\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y}\right)
= -\frac{\partial p}{\partial x}
+ \frac{\partial}{\partial x}\!\left(\frac{4\mu}{3}\frac{\partial u}{\partial x}
- \frac{2\mu}{3}\frac{\partial v}{\partial y}\right)
+ \frac{\partial}{\partial y}\!\left(\mu\frac{\partial u}{\partial y}
+ \mu\frac{\partial v}{\partial x}\right)
\]
y-momentum
\[
\rho \left(\frac{\partial v}{\partial t} + u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y}\right)
= -\frac{\partial p}{\partial y}
+ \frac{\partial}{\partial x}\!\left(\mu\frac{\partial v}{\partial x}
+ \mu\frac{\partial u}{\partial y}\right)
+ \frac{\partial}{\partial y}\!\left(\frac{4\mu}{3}\frac{\partial v}{\partial y}
- \frac{2\mu}{3}\frac{\partial u}{\partial x}\right)
\]
Energy (in terms of temperature)
\[
\rho c_p \left(\frac{\partial T}{\partial t} + u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y}\right)
= \left(\frac{\partial p}{\partial t} + u\frac{\partial p}{\partial x} + v\frac{\partial p}{\partial y}\right)
+ \frac{\partial}{\partial x}\!\left(k\frac{\partial T}{\partial x}\right)
+ \frac{\partial}{\partial y}\!\left(k\frac{\partial T}{\partial y}\right)
+ \Phi
\]
where
\[
\Phi = \mu\left[2\left(\frac{\partial u}{\partial x}\right)^2 + 2\left(\frac{\partial v}{\partial y}\right)^2 + \left(\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}\right)^2 - \frac{2}{3}\left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}\right)^2\right]
\]
Steady Flow
For steady flow, all \(\partial/\partial t\) terms vanish. The continuity, momentum, and energy equations reduce to:
\[
\frac{\partial (\rho u)}{\partial x} + \frac{\partial (\rho v)}{\partial y} = 0
\]
\[
\rho \left(u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y}\right)
= -\frac{\partial p}{\partial x}
+ \frac{\partial}{\partial x}\!\left(\frac{4\mu}{3}\frac{\partial u}{\partial x}
- \frac{2\mu}{3}\frac{\partial v}{\partial y}\right)
+ \frac{\partial}{\partial y}\!\left(\mu\frac{\partial u}{\partial y}
+ \mu\frac{\partial v}{\partial x}\right)
\]
\[
\rho \left(u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y}\right)
= -\frac{\partial p}{\partial y}
+ \frac{\partial}{\partial x}\!\left(\mu\frac{\partial v}{\partial x}
+ \mu\frac{\partial u}{\partial y}\right)
+ \frac{\partial}{\partial y}\!\left(\frac{4\mu}{3}\frac{\partial v}{\partial y}
- \frac{2\mu}{3}\frac{\partial u}{\partial x}\right)
\]
\[
\rho c_p \left(u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y}\right)
= u\frac{\partial p}{\partial x} + v\frac{\partial p}{\partial y}
+ \frac{\partial}{\partial x}\!\left(k\frac{\partial T}{\partial x}\right)
+ \frac{\partial}{\partial y}\!\left(k\frac{\partial T}{\partial y}\right)
+ \Phi
\]
Boundary Layer Scaling
The steady NS equations above simplify when the boundary layer is thin relative to the streamwise length scale. Introduce a reference length \(L\) and free-stream speed \(U_\infty\), and let \(\delta \ll L\) be the BL thickness.
Step 1: get \(V\) scaling from continuity. Both terms of the continuity equation must be the same order:
\[
\frac{U_\infty}{L} \sim \frac{V}{\delta}
\qquad\Longrightarrow\qquad
V \sim \frac{\delta}{L}\,U_\infty
\]
Step 2: get \(\delta\) by balancing convection with wall-normal viscosity in x-momentum.
\[
\underbrace{\rho\,\frac{U_\infty^2}{L}}_{\text{inertia}} \sim \underbrace{\frac{\mu\,U_\infty}{\delta^2}}_{\text{viscosity}}
\qquad\Longrightarrow\qquad
\delta \sim \frac{L}{\sqrt{Re_L}}, \qquad Re_L = \frac{\rho U_\infty L}{\mu}
\]
Step 3: Define dimensionless variables
\[
\begin{aligned}
x^* &= \frac{x}{L}, &\quad
y^* &= \frac{y}{\delta}, &\quad
u^* &= \frac{u}{U_\infty}, \\[6pt]
v^* &= \frac{v}{\varepsilon U_\infty}, &\quad
p^* &= \frac{p}{\rho_\infty U_\infty^2}, &\quad
\rho^* &= \frac{\rho}{\rho_\infty}, \\[6pt]
\mu^* &= \frac{\mu}{\mu_\infty}, &\quad
T^* &= \frac{T}{T_\infty}, &\quad
k^* &= \frac{k}{k_\infty}
\end{aligned}
\]
where \(\varepsilon = \delta/L = Re_L^{-1/2} \ll 1\).
Step 4: Substitute
Using \(\partial/\partial x = (1/L)\,\partial/\partial x^*\) and \(\partial/\partial y = (1/\delta)\,\partial/\partial y^*\), substitute into each equation and divide out the common dimensional factor. Terms small by \(\varepsilon\) or \(\varepsilon^2\) appear with an explicit prefactor.
Continuity
\[
\cancel{\frac{\rho_\infty U_\infty}{L}}\,\frac{\partial(\rho^* u^*)}{\partial x^*}
+ \cancel{\frac{\rho_\infty U_\infty}{L}}\,\frac{\partial(\rho^* v^*)}{\partial y^*} = 0
\]
\[
\frac{\partial(\rho^* u^*)}{\partial x^*} + \frac{\partial(\rho^* v^*)}{\partial y^*} = 0
\]
x-momentum
\[
\begin{aligned}
&\cancel{\frac{\rho_\infty U_\infty^2}{L}}\rho^*\!\left(u^*\frac{\partial u^*}{\partial x^*} + v^*\frac{\partial u^*}{\partial y^*}\right)
= -\cancel{\frac{\rho_\infty U_\infty^2}{L}}\frac{\partial p^*}{\partial x^*}
+ \cancel{\frac{\rho_\infty U_\infty^2}{L}}\frac{\partial}{\partial y^*}\!\left(\mu^*\frac{\partial u^*}{\partial y^*}\right) \\[6pt]
&\quad+ \varepsilon^2\cancel{\frac{\rho_\infty U_\infty^2}{L}}\!\left[
\frac{\partial}{\partial x^*}\!\left(\frac{4\mu^*}{3}\frac{\partial u^*}{\partial x^*}
- \frac{2\mu^*}{3}\frac{\partial v^*}{\partial y^*}\right)
+ \frac{\partial}{\partial y^*}\!\left(\mu^*\frac{\partial v^*}{\partial x^*}\right)
\right]
\end{aligned}
\]
\[
\begin{aligned}
\rho^*\!\left(u^*\frac{\partial u^*}{\partial x^*} + v^*\frac{\partial u^*}{\partial y^*}\right)
&= -\frac{\partial p^*}{\partial x^*}
+ \frac{\partial}{\partial y^*}\!\left(\mu^*\frac{\partial u^*}{\partial y^*}\right) \\[6pt]
&+ \varepsilon^2\!\left[
\frac{\partial}{\partial x^*}\!\left(\frac{4\mu^*}{3}\frac{\partial u^*}{\partial x^*}
- \frac{2\mu^*}{3}\frac{\partial v^*}{\partial y^*}\right)
+ \frac{\partial}{\partial y^*}\!\left(\mu^*\frac{\partial v^*}{\partial x^*}\right)
\right]
\end{aligned}
\]
y-momentum
\[
\begin{gathered}
\varepsilon\cancel{\frac{\rho_\infty U_\infty^2}{L}}\rho^*\!\left(u^*\frac{\partial v^*}{\partial x^*} + v^*\frac{\partial v^*}{\partial y^*}\right)
= -\cancel{\frac{\rho_\infty U_\infty^2}{L}}\frac{1}{\varepsilon}\frac{\partial p^*}{\partial y^*} \\[6pt]
+ \varepsilon\cancel{\frac{\rho_\infty U_\infty^2}{L}}\!\left[
\frac{\partial}{\partial x^*}\!\left(\mu^*\frac{\partial v^*}{\partial x^*} + \mu^*\frac{\partial u^*}{\partial y^*}\right)
+ \frac{\partial}{\partial y^*}\!\left(\frac{4\mu^*}{3}\frac{\partial v^*}{\partial y^*}
- \frac{2\mu^*}{3}\frac{\partial u^*}{\partial x^*}\right)
\right]
\end{gathered}
\]
Multiplying through by \(\varepsilon\):
\[
\begin{aligned}
\varepsilon^2\,\rho^*\!\left(u^*\frac{\partial v^*}{\partial x^*} + v^*\frac{\partial v^*}{\partial y^*}\right)
&= -\frac{\partial p^*}{\partial y^*} \\[6pt]
&+ \varepsilon^2\!\left[
\frac{\partial}{\partial x^*}\!\left(\mu^*\frac{\partial v^*}{\partial x^*} + \mu^*\frac{\partial u^*}{\partial y^*}\right)
+ \frac{\partial}{\partial y^*}\!\left(\frac{4\mu^*}{3}\frac{\partial v^*}{\partial y^*}
- \frac{2\mu^*}{3}\frac{\partial u^*}{\partial x^*}\right)
\right]
\end{aligned}
\]
Energy
\[
\begin{aligned}
&\cancel{\frac{\rho_\infty c_p T_\infty U_\infty}{L}}\rho^*\!\left(u^*\frac{\partial T^*}{\partial x^*} + v^*\frac{\partial T^*}{\partial y^*}\right) \\[6pt]
&\quad= \cancel{\frac{\rho_\infty c_p T_\infty U_\infty}{L}}\mathrm{Ec}\,u^*\frac{\partial p^*}{\partial x^*}
+ \cancel{\frac{\rho_\infty c_p T_\infty U_\infty}{L}}\frac{1}{\mathrm{Pr}}\frac{\partial}{\partial y^*}\!\left(k^*\frac{\partial T^*}{\partial y^*}\right)
+ \cancel{\frac{\rho_\infty c_p T_\infty U_\infty}{L}}\frac{\mathrm{Ec}}{\mathrm{Pr}}\,\mu^*\!\left(\frac{\partial u^*}{\partial y^*}\right)^{\!2} \\[6pt]
&\quad+ \varepsilon^2\cancel{\frac{\rho_\infty c_p T_\infty U_\infty}{L}}\frac{1}{\mathrm{Pr}}\frac{\partial}{\partial x^*}\!\left(k^*\frac{\partial T^*}{\partial x^*}\right) \\[6pt]
&\quad+ \varepsilon^2\cancel{\frac{\rho_\infty c_p T_\infty U_\infty}{L}}\frac{\mathrm{Ec}}{\mathrm{Pr}}\,\mu^*\!\left[
2\!\left(\frac{\partial u^*}{\partial x^*}\right)^{\!2}
+ 2\!\left(\frac{\partial v^*}{\partial y^*}\right)^{\!2}
+ \!\left(\frac{\partial v^*}{\partial x^*}\right)^{\!2}
- \frac{2}{3}\!\left(\frac{\partial u^*}{\partial x^*} + \frac{\partial v^*}{\partial y^*}\right)^{\!2}
\right]
\end{aligned}
\]
where \(\mathrm{Pr} = \mu_\infty c_p / k_\infty\) and \(\mathrm{Ec} = U_\infty^2/(c_p T_\infty)\).
\[
\begin{aligned}
\rho^*\!\left(u^*\frac{\partial T^*}{\partial x^*} + v^*\frac{\partial T^*}{\partial y^*}\right)
&= \mathrm{Ec}\,u^*\frac{\partial p^*}{\partial x^*}
+ \frac{1}{\mathrm{Pr}}\frac{\partial}{\partial y^*}\!\left(k^*\frac{\partial T^*}{\partial y^*}\right)
+ \frac{\mathrm{Ec}}{\mathrm{Pr}}\,\mu^*\!\left(\frac{\partial u^*}{\partial y^*}\right)^{\!2} \\[6pt]
&+ \varepsilon^2\frac{1}{\mathrm{Pr}}\frac{\partial}{\partial x^*}\!\left(k^*\frac{\partial T^*}{\partial x^*}\right) \\[6pt]
&+ \varepsilon^2\frac{\mathrm{Ec}}{\mathrm{Pr}}\,\mu^*\!\left[
2\!\left(\frac{\partial u^*}{\partial x^*}\right)^{\!2}
+ 2\!\left(\frac{\partial v^*}{\partial y^*}\right)^{\!2}
+ \!\left(\frac{\partial v^*}{\partial x^*}\right)^{\!2}
- \frac{2}{3}\!\left(\frac{\partial u^*}{\partial x^*} + \frac{\partial v^*}{\partial y^*}\right)^{\!2}
\right]
\end{aligned}
\]
Step 5: drop \(\mathcal{O}(\varepsilon^2)\) terms.
Setting \(\varepsilon \to 0\) in each equation above:
\[
\frac{\partial(\rho^* u^*)}{\partial x^*} + \frac{\partial(\rho^* v^*)}{\partial y^*} = 0
\]
\[
\rho^*\!\left(u^*\frac{\partial u^*}{\partial x^*} + v^*\frac{\partial u^*}{\partial y^*}\right)
= -\frac{\partial p^*}{\partial x^*}
+ \frac{\partial}{\partial y^*}\!\left(\mu^*\frac{\partial u^*}{\partial y^*}\right)
\]
\[\frac{\partial p^*}{\partial y^*} = 0\]
\[
\rho^*\!\left(u^*\frac{\partial T^*}{\partial x^*} + v^*\frac{\partial T^*}{\partial y^*}\right)
= \mathrm{Ec}\,u^*\frac{\partial p^*}{\partial x^*}
+ \frac{1}{\mathrm{Pr}}\frac{\partial}{\partial y^*}\!\left(k^*\frac{\partial T^*}{\partial y^*}\right)
+ \frac{\mathrm{Ec}}{\mathrm{Pr}}\,\mu^*\!\left(\frac{\partial u^*}{\partial y^*}\right)^{\!2}
\]
Step 6: re-dimensionalize.
Reversing the substitutions (\(u^* \to u/U_\infty\), \(y^* \to y/\delta\), \(p^* \to p/(\rho_\infty U_\infty^2)\), etc.) and applying \(\partial p^*/\partial y^* = 0 \Rightarrow \partial p/\partial y = 0\) recovers the dimensional BL equations shown at the top of this page.
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White, F. M. (2006). Viscous Fluid Flow, 3rd ed. McGraw-Hill, New York. ↩
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Schlichting, H. & Gersten, K. (2017). Boundary Layer Theory, 9th ed. Springer. DOI: 10.1007/978-3-662-52919-5 ↩