Derivation
Note
This derivation uses the Levy-Lees transformation and assumes \(\rho_e\mu_e =
\text{const}\) . For the more general derivation using the
Illingworth-Stewartson transformation, see Derivation (IS) .
Starting from the steady 2D compressible BL equations :
\[
\frac{\partial(\rho u)}{\partial x} + \frac{\partial(\rho v)}{\partial y} = 0
\]
\[
\rho\!\left(u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y}\right)
= -\frac{dp}{dx} + \frac{\partial}{\partial y}\!\left(\mu\frac{\partial u}{\partial y}\right)
\]
\[
\rho c_p\!\left(u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y}\right)
= u\frac{dp}{dx} + \frac{\partial}{\partial y}\!\left(k\frac{\partial T}{\partial y}\right)
+ \mu\!\left(\frac{\partial u}{\partial y}\right)^{\!2}
\]
Similarity Ansatz
Edge velocity power law :
\[u_e(x) = C x^m\]
Dimensionless stream function :
\[\psi = \sqrt{2\xi}\,f(\eta)\]
Outer flow
The outer inviscid flow satisfies the Euler x-momentum equation (see
2D BL equations ):
\[-\frac{dp}{dx} = \rho_e u_e \frac{du_e}{dx}\]
For the power-law edge velocity \(u_e = Cx^m\) :
\[
\frac{du_e}{dx} = \frac{d}{dx}\left(Cx^{m}\right)
= m C x^{m-1} = \frac{m}{x} \underbrace{C x^m}_{u_e} = \frac{m u_e}{x}
\]
so the pressure gradient becomes:
\[-\frac{dp}{dx} = \rho_e u_e^2\frac{m}{x}\]
Definitions
Stream function (compressible form):
\[\rho u = \frac{\partial\psi}{\partial y}, \qquad \rho v = -\frac{\partial\psi}{\partial x}\]
Levy-Lees similarity coordinates :
\[\xi = \int_0^x \rho_e\mu_e u_e\,dx', \qquad \eta = \frac{u_e}{\sqrt{2\xi}}\int_0^y\rho\,dy'\]
For the power-law \(u_e = Cx^m\) , the edge quantities \(\rho_e\) and \(\mu_e\) are constant
(isentropic edge, uniform composition), so \(\rho_e\mu_e\) can be taken out of the integral:
\[
\xi = \rho_e\mu_e\int_0^x C x'^m\,dx'
= \rho_e\mu_e\,C\,\frac{x^{m+1}}{m+1}
= \frac{\rho_e\mu_e\,\overbrace{Cx^m}^{u_e}\,x}{m+1}
= \frac{\rho_e\mu_e u_e x}{m+1}
\]
Hartree parameter :
\[\beta_H = \frac{2m}{m+1}\]
Chapman-Rubesin factor :
\[C = \frac{\rho\mu}{\rho_e\mu_e}\]
temperature ratio :
\[
\tau = \frac{T}{T_e}
\]
Ideal gas law :
From the ideal gas law \(p = \rho R T\) , the density is \(\rho = p/(RT)\) , and at the boundary layer edge \(\rho_e = p_e/(RT_e)\) .
From the y-momentum boundary layer equation
\[
\frac{\partial p}{\partial y} = 0 \rightarrow p = p_e
\]
Therefore:
\[
\frac{\rho}{\rho_e}
= \frac{p/(RT)}{p_e/(RT_e)}
= \frac{T_e}{T}
= \frac{1}{\tau}
\]
Edge Mach number :
\[
M_e = \frac{u_e}{\sqrt{\gamma R T_e}}
\]
With \(c_p = \gamma R/(\gamma-1)\) it follows that
\[
\frac{u_e^2}{c_p T_e} = (\gamma-1)M_e^2
\]
Prandtl number :
\[\mathrm{Pr} = \frac{\mu c_p}{k}\]
so that \(k = \mu c_p/\mathrm{Pr}\) .
Partial derivatives of the similarity coordinates
To transform any \(\partial/\partial x\big|_y\) or \(\partial/\partial y\big|_x\) term, we need
the partial derivatives of \(\xi\) and \(\eta\) with respect to \(x\) and \(y\) . Both follow from the
Leibniz integral rule :
\[\frac{d}{dx}\!\left(\int_{a(x)}^{b(x)} f(x,t)\,dt\right)
= f(x,b(x))\frac{d b}{dx} - f(x,a(x))\frac{d a}{dx}
+ \int_{a(x)}^{b(x)}\frac{\partial f}{\partial x}\,dt\]
For \(\xi\) : the lower limit is constant (\(a=0\) ), the upper limit is \(b=x\) , and the integrand
\(\rho_e\mu_e u_e\) depends only on the dummy variable \(x'\) (not on \(x\) explicitly):
\[
\frac{\partial\xi}{\partial x} = \frac{\partial}{\partial x}\left(\int_0^x \rho_e\mu_e u_e\,dx'\right) = \rho_e\mu_e u_e
\]
For \(\eta\) : differentiating with respect to \(y\) , the lower limit is constant (\(a=0\) ), the upper
limit is \(b=y\) , and the integrand \(\rho\) depends only on \(y'\) :
\[
\frac{\partial\eta}{\partial y} = \frac{u_e}{\sqrt{2\xi}}\frac{\partial}{\partial y}\left(\int_0^y \rho\,dy'\right) = \frac{\rho u_e}{\sqrt{2\xi}}
\]
\[\frac{\partial\eta}{\partial x}\bigg|_y = \frac{\partial}{\partial x}\!\left(\frac{u_e}{\sqrt{2\xi}}\int_0^y\rho\,dy'\right)\]
The change of variables is \((x, y) \to (\xi, \eta)\) . For any function \(F(\xi, \eta)\) , the
chain rule gives:
\[\frac{\partial F}{\partial y}\bigg|_x
= \frac{\partial F}{\partial \xi}\bigg|_\eta \underbrace{\frac{\partial \xi}{\partial y}\bigg|_x}_{=\,0}
+ \frac{\partial F}{\partial \eta}\bigg|_\xi \frac{\partial \eta}{\partial y}\bigg|_x
= \frac{\rho u_e}{\sqrt{2\xi}}\frac{\partial F}{\partial \eta}\bigg|_\xi\]
The \(\partial\xi/\partial y\big|_x = 0\) because \(\xi = \int_0^x \rho_e\mu_e u_e\,dx'\) contains
no \(y\) -dependence.
\[\frac{\partial F}{\partial x}\bigg|_y
= \frac{\partial F}{\partial \xi}\bigg|_\eta \frac{\partial \xi}{\partial x}\bigg|_y
+ \frac{\partial F}{\partial \eta}\bigg|_\xi \frac{\partial \eta}{\partial x}\bigg|_y\]
Streamwise velocity
From \(\rho u = \partial\psi/\partial y\) and the \(\partial/\partial y\) operator:
\[
\rho u = \frac{\partial\psi}{\partial y}
= \frac{\partial\overbrace{\psi}^{\sqrt{2\xi} f(\eta)}}{\partial \eta} \overbrace{\frac{\partial \eta}{\partial y}}^{\frac{\rho u_e}{\sqrt{2\xi}}}
= \sqrt{2\xi} \overbrace{\frac{\partial f(\eta)}{\partial \eta}}^{f'(\eta)} \frac{\rho u_e}{\sqrt{2\xi}}
= \rho\,u_e f'(\eta)
\qquad\Longrightarrow\qquad u = u_e f'(\eta)
\]
The streamwise derivative of \(u\) follows the chain rule on \(u = u_e(\xi)\,f'(\eta)\) .
Since \(u_e\) depends on \(x\) only through \(\xi\) , the two \(u_e\) chain-rule factors collapse
immediately: \((\partial u_e/\partial\xi)\,(\partial\xi/\partial x) = du_e/dx\) :
\[
\frac{\partial \overbrace{u}^{u_e f'(\eta)}}{\partial x}\bigg|_y
= \overbrace{\frac{du_e}{dx}}^{mu_e/x} f'(\eta)
+ u_e\,\overbrace{\frac{\partial f'(\eta)}{\partial \eta}\bigg|_\xi}^{f''(\eta)}\,
\frac{\partial \eta}{\partial x}\bigg|_y
= \frac{m u_e}{x}\,f'(\eta) + u_e\,f''(\eta)\,\frac{\partial\eta}{\partial x}\bigg|_y
\]
The wall-normal derivative of \(u\) :
\[
\frac{\partial \overbrace{u}^{u_e f'(\eta)}}{\partial y}
= \frac{\partial \left(u_e f'(\eta)\right)}{\partial\eta} \overbrace{\frac{\partial\eta}{\partial y}}^{\frac{\rho u_e}{\sqrt{2 \xi}}}
= \frac{\rho u_e}{\sqrt{2\xi}}\frac{\partial(u_e f'(\eta))}{\partial\eta}
= \frac{\rho u_e^2}{\sqrt{2\xi}}f''(\eta)
\]
Continuity
\[
\frac{\partial(\overbrace{\rho u}^{\frac{\partial\psi}{\partial y}})}{\partial x}
+ \frac{\partial(\overbrace{\rho v}^{-\frac{\partial\psi}{\partial x}})}{\partial y}
=
\frac{\partial^2\psi}{\partial x\,\partial y}
- \frac{\partial^2\psi}{\partial y\,\partial x} = 0
\]
Continuity is satisfied identically.
x-momentum
Using the expression derived above the transformed terms are:
\(\rho u\frac{\partial u}{\partial x}\) term.
\[\rho u\frac{\partial u}{\partial x} = \rho u_e f' \left(\frac{m u_e}{x}f' + u_e f'' \frac{\partial\eta}{\partial x}\right)
= \frac{\rho u_e^2}{x}\!\left(m f'^2 + x f'' f'\frac{\partial\eta}{\partial x}\right)\]
\(\rho v \frac{\partial u}{\partial y}\) term.\(\rho v = -\partial\psi/\partial x\)
by applying the product rule to \(\psi = \sqrt{2\xi}\,f(\eta)\) :
\[
\rho v = -\frac{\partial \overbrace{\psi}^{\sqrt{2\xi}\,f(\eta)}}{\partial x}\bigg|_y
= -\left[
\overbrace{\frac{\partial\sqrt{2\xi}}{\partial x}}^{\rho_e\mu_e u_e/\sqrt{2\xi}}\,f
+ \sqrt{2\xi}\;\overbrace{\frac{\partial f(\eta)}{\partial x}\bigg|_y}^{f'\,\partial\eta/\partial x\big|_y}
\right]
= -\frac{\rho_e\mu_e u_e}{\sqrt{2\xi}}\,f - \sqrt{2\xi}\,f'\,\frac{\partial\eta}{\partial x}\bigg|_y
\]
Multiplying by \(\partial u/\partial y = \rho u_e^2 f''/\sqrt{2\xi}\) :
\[
\rho v\,\frac{\partial u}{\partial y}
= \left(-\frac{\rho_e\mu_e u_e}{\sqrt{2\xi}}\,f - \sqrt{2\xi}\,f'\,\frac{\partial\eta}{\partial x}\bigg|_y\right)
\frac{\rho u_e^2}{\sqrt{2\xi}}\,f''
= -\frac{\rho\rho_e\mu_e u_e^3}{2\xi}\,ff''
- \rho u_e^2\,f'f''\,\frac{\partial\eta}{\partial x}\bigg|_y
\]
Combined convective term. Adding both terms the \(\partial\eta/\partial x\) pieces cancel:
\[
\rho u\,\frac{\partial u}{\partial x} + \rho v\,\frac{\partial u}{\partial y}
= \frac{m\rho u_e^2}{x}\,f'^2
+ \cancel{\rho u_e^2\,f'f''\,\frac{\partial\eta}{\partial x}}
- \frac{\rho\rho_e\mu_e u_e^3}{2\xi}\,ff''
- \cancel{\rho u_e^2\,f'f''\,\frac{\partial\eta}{\partial x}}
= \frac{m\rho u_e^2}{x}\,f'^2 - \frac{\rho\rho_e\mu_e u_e^3}{2\xi}\,ff''
\]
Recast \(\frac{m}{x}\) :
\[
\frac{m}{\underbrace{x}_{\frac{\xi (m+1)}{\rho_e \mu_e u_e}}}
= \frac{m\rho_e\mu_e u_e}{\xi(m+1)}
= \overbrace{\frac{m}{(m+1)}}^{\beta_H/2} \frac{\rho_e \mu_e u_e}{\xi}
= \frac{\beta_H\rho_e\mu_e u_e}{2\xi}
\]
Substituting:
\[
\rho u\,\frac{\partial u}{\partial x} + \rho v\,\frac{\partial u}{\partial y}
= \overbrace{\frac{m}{x}}^{\frac{\beta_H\rho_e\mu_e u_e}{2\xi}} \rho u_e^2 f'^2
- \frac{\rho\rho_e\mu_e u_e^3}{2\xi}\,ff''
= \frac{\beta_H\rho\rho_e\mu_e u_e^3}{2\xi} f'^2 - \frac{\rho\rho_e\mu_e u_e^3}{2\xi}\,ff''
\]
\[\boxed{
\rho u\,\frac{\partial u}{\partial x} + \rho v\,\frac{\partial u}{\partial y}
= \frac{\rho\rho_e\mu_e u_e^3}{2\xi}\!\left(\beta_H f'^2 - ff''\right)
}\]
Pressure term.
\[
-\frac{dp}{dx}
= \rho_e u_e^2\frac{m}{x}
= \rho_e u_e^2 \frac{\beta_H \rho_e \mu_e u_e}{2 \xi}
= \overbrace{\rho_e}^{\rho \tau} \frac{\rho_e \mu_e u_e^3}{2\xi} \beta_H
= \frac{\rho\rho_e\mu_e u_e^3}{2\xi} \tau \beta_H
\]
\[\boxed{-\frac{dp}{dx} = \frac{\rho\rho_e\mu_e u_e^3}{2\xi} \tau \beta_H}\]
Viscous term.
\[
\mu\frac{\partial \overbrace{u}^{u_e f'}}{\partial y}
= \overbrace{\mu}^{C\rho_e\mu_e/\rho} \cdot \overbrace{\frac{\partial u}{\partial y}}^{\rho u_e^2 f''/\sqrt{2\xi}}
= \frac{C\rho_e\mu_e u_e^2}{\sqrt{2\xi}}\,f''
\]
Now apply the outer \(\partial/\partial y = (\rho u_e/\sqrt{2\xi})\,\partial/\partial\eta\) ,
noting that \(\rho_e\mu_e u_e^2/\sqrt{2\xi}\) does not depend on \(\eta\) :
\[
\frac{\partial}{\partial y}\!\left(\mu\frac{\partial u}{\partial y}\right)
= \overbrace{\frac{\partial}{\partial y}}^{\frac{\rho u_e}{\sqrt{2\xi}}\partial/\partial\eta}
\left(\overbrace{\mu\frac{\partial u}{\partial y}}^{C\rho_e\mu_e u_e^2 f''/\sqrt{2\xi}}\right)
= \frac{\rho u_e}{\sqrt{2\xi}}\cdot\frac{\rho_e\mu_e u_e^2}{\sqrt{2\xi}}\,(Cf'')'
\]
\[\boxed{\frac{\partial}{\partial y}\!\left(\mu\frac{\partial u}{\partial y}\right)
= \frac{\rho\rho_e\mu_e u_e^3}{2\xi}\,(Cf'')'}\]
Assembly.
The x-momentum equation \(\rho u\,\partial u/\partial x + \rho v\,\partial u/\partial y = -dp/dx + \partial(\mu\,\partial u/\partial y)/\partial y\) becomes:
\[
\cancel{\frac{\rho\rho_e\mu_e u_e^3}{2\xi}}\!\left(\beta_H f'^2 - ff''\right)
= \cancel{\frac{\rho\rho_e\mu_e u_e^3}{2\xi}}\beta_H\tau
+ \cancel{\frac{\rho\rho_e\mu_e u_e^3}{2\xi}}\,(Cf'')'
\]
Dividing through by \(\rho\rho_e\mu_e u_e^3/(2\xi)\) and rearranging:
\[(Cf'')' + ff'' + \beta_H(\tau - f'^2) = 0\]
Energy
The energy equation is:
\[
\rho c_p\!\left(u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y}\right)
= u\frac{dp}{dx}
+ \frac{\partial}{\partial y}\!\left(k\frac{\partial T}{\partial y}\right)
+ \mu\!\left(\frac{\partial u}{\partial y}\right)^{\!2}
\]
Note \(T = T_e\,\tau(\eta) \rightarrow\) function of \(\eta\) only
\[
\frac{\partial \overbrace{T}^{T_e\,\tau(\eta)}}{\partial y}
= T_e\,\overbrace{\frac{\partial\tau}{\partial\eta}}^{\tau'}\,
\overbrace{\frac{\partial\eta}{\partial y}}^{\rho u_e/\sqrt{2\xi}}
= \frac{\rho u_e T_e}{\sqrt{2\xi}}\,\tau'
\]
\[
\frac{\partial \overbrace{T}^{T_e\,\tau(\eta)}}{\partial x}\bigg|_y
= T_e\,\overbrace{\frac{\partial\tau}{\partial\eta}}^{\tau'}\,\frac{\partial\eta}{\partial x}\bigg|_y
= T_e\,\tau'\,\frac{\partial\eta}{\partial x}\bigg|_y
\]
Convective term.
Streamwise :
\[
\rho c_p u\,\partial T/\partial x = \rho c_p u_e f' \cdot T_e\tau'\,\partial\eta/\partial x\big|_y
\]
Wall-normal : using \(\rho v\) from the x-momentum section:
\[
c_p(\rho v)\frac{\partial T}{\partial y}
= c_p\!\left(-\frac{\rho_e\mu_e u_e}{\sqrt{2\xi}}\,f
- \sqrt{2\xi}\,f'\,\frac{\partial\eta}{\partial x}\bigg|_y\right)
\frac{\rho u_e T_e\,\tau'}{\sqrt{2\xi}}
= -\frac{\rho\rho_e\mu_e u_e^2 c_p T_e}{2\xi}\,f\tau'
- \rho u_e c_p T_e f'\tau'\,\frac{\partial\eta}{\partial x}\bigg|_y
\]
Combining the above terms
\[
\rho c_p\!\left(u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y}\right)
= \cancel{\rho c_p u_e T_e f'\tau'\frac{\partial\eta}{\partial x}}
- \frac{\rho\rho_e\mu_e u_e^2 c_p T_e}{2\xi}\,f\tau'
- \cancel{\rho c_p u_e T_e f'\tau'\frac{\partial\eta}{\partial x}}
\]
\[\boxed{\rho c_p\!\left(u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y}\right)
= -\frac{\rho\rho_e\mu_e u_e^2 c_p T_e}{2\xi}\,f\tau'}\]
Diffusion term.
\[
k\frac{\partial T}{\partial y}
= \overbrace{k}^{C\rho_e\mu_e c_p/(\rho\,\mathrm{Pr})} \cdot
\overbrace{\frac{\partial T}{\partial y}}^{\rho u_e T_e\tau'/\sqrt{2\xi}}
= \frac{C\rho_e\mu_e u_e c_p T_e}{\mathrm{Pr}\sqrt{2\xi}}\,\tau'
\]
Applying the outer \(\partial/\partial y = (\rho u_e/\sqrt{2\xi})\,\partial/\partial\eta\) :
\[
\frac{\partial}{\partial y}\!\left(k\frac{\partial T}{\partial y}\right)
= \overbrace{\frac{\partial}{\partial y}}^{\frac{\rho u_e}{\sqrt{2\xi}}\partial/\partial\eta}
\left(\overbrace{k\frac{\partial T}{\partial y}}^{C\rho_e\mu_e u_e c_p T_e\tau'/(\mathrm{Pr}\sqrt{2\xi})}\right)
= \frac{\rho u_e}{\sqrt{2\xi}}\cdot\frac{\rho_e\mu_e u_e c_p T_e}{\mathrm{Pr}\sqrt{2\xi}}\,(C\tau')'
\]
\[\boxed{\frac{\partial}{\partial y}\!\left(k\frac{\partial T}{\partial y}\right)
= \frac{\rho\rho_e\mu_e u_e^2 c_p T_e}{2\xi}\left(\frac{C}{\mathrm{Pr}}\tau'\right)'}\]
Pressure work term.
\[
u\frac{dp}{dx}
= \overbrace{u_e f'}^{u}\cdot\left(-\rho_e u_e^2\overbrace{\frac{m}{x}}^{\beta_H\rho_e\mu_e u_e/(2\xi)}\right)
= -\frac{\beta_H\rho_e^2\mu_e u_e^4 f'}{2\xi}
= -\frac{\rho\rho_e\mu_e u_e^2 c_p T_e}{2\xi}\cdot
\overbrace{\frac{\beta_H\tau u_e^2 f'}{c_p T_e}}^{(\gamma-1)M_e^2\beta_H\tau f'}
\]
\[\boxed{u\frac{dp}{dx}
= -\frac{\rho\rho_e\mu_e u_e^2 c_p T_e}{2\xi}\,(\gamma-1)M_e^2\,\beta_H\tau f'}\]
Dissipation term.
\[
\mu\!\left(\frac{\partial u}{\partial y}\right)^{\!2}
= \overbrace{\frac{C\rho_e\mu_e}{\rho}}^{\mu}
\left(\overbrace{\frac{\rho u_e^2 f''}{\sqrt{2\xi}}}^{\partial u/\partial y}\right)^{\!2}
= \frac{C\rho\rho_e\mu_e u_e^4 f''^2}{2\xi}
= \frac{\rho\rho_e\mu_e u_e^2 c_p T_e}{2\xi}\cdot
\overbrace{\frac{C u_e^2 f''^2}{c_p T_e}}^{(\gamma-1)M_e^2\,Cf''^2}
\]
\[\boxed{\mu\!\left(\frac{\partial u}{\partial y}\right)^{\!2}
= \frac{\rho\rho_e\mu_e u_e^2 c_p T_e}{2\xi}\,(\gamma-1)M_e^2\,Cf''^2}\]
Assembly. Every term carries \(\rho\rho_e\mu_e u_e^2 c_p T_e/(2\xi)\) :
\[
-\cancel{\frac{\rho\rho_e\mu_e u_e^2 c_p T_e}{2\xi}}\,f\tau'
= -\cancel{\frac{\rho\rho_e\mu_e u_e^2 c_p T_e}{2\xi}}\,(\gamma-1)M_e^2\beta_H\tau f'
+ \cancel{\frac{\rho\rho_e\mu_e u_e^2 c_p T_e}{2\xi}}\left(\frac{C}{\mathrm{Pr}}\tau'\right)'
+ \cancel{\frac{\rho\rho_e\mu_e u_e^2 c_p T_e}{2\xi}}\,(\gamma-1)M_e^2 Cf''^2
\]
Dividing through and rearranging:
\[\left(\frac{C}{\mathrm{Pr}}\tau'\right)' + f\tau'
+ (\gamma-1)M_e^2\!\left[Cf''^2 - \beta_H\tau f'\right] = 0\]