Boundary layer

Contenuti

9. Boundary layer#

9.1. Equazioni di Prandtl dello strato limite#

Steady Navier—Stokes equations for a 2-dimensional flow using Cartesian coordinates read

\[\begin{split}\begin{cases} u \partial_x u + v \partial_y u - \nu \left( \partial_{xx} u + \partial_{yy} u \right) + \frac{1}{\rho} \partial_x P = 0 \\ u \partial_x v + v \partial_y v - \nu \left( \partial_{xx} v + \partial_{yy} v \right) + \frac{1}{\rho} \partial_y P = 0 \\ \partial_x u + \partial_y v = 0 \end{cases}\end{split}\]

Let’s assume the flow of interest has a main stream-wise direction, here identified by \(x\) coordinate, while \(y\) runs in the orthogonal direction. Let \(U\), \(V\), \(X\), \(\delta\) be characteristic \(x\)- and \(y\)- components of the velocity field and lengths. Non-dimensional equations become

\[\begin{split}\begin{cases} \frac{U^2}{X} u \partial_x u + \frac{U V}{\delta} v \partial_y u - \frac{\nu U}{X^2} \partial_{xx} u - \frac{\nu U}{\delta^2} \partial_{yy} u + \frac{P}{\rho X} \partial_x P = 0 \\ \frac{U V}{X} u \partial_x v + \frac{V^2}{\delta} v \partial_y v - \frac{\nu V}{X^2} \partial_{xx} v - \frac{\nu V}{\delta^2} \partial_{yy} v + \frac{P}{\rho \delta} \partial_y P = 0 \\ \frac{U}{X} \partial_x u + \frac{V}{\delta} \partial_y v = 0 \end{cases}\end{split}\]

Let the 2 terms in the incompressibility constraint have the same order of magnitude. It follows that \(V \sim \frac{\delta}{X} U\). Using this relation, and assuming \(P \sim \rho U^2\), the non-dimensional equations become

\[\begin{split}\begin{cases} \frac{U^2}{X} \left[ \left( u \partial_x u + v \partial_y u - \partial_{yy} u + \partial_x P \right) - \left(\frac{\delta}{X}\right)^2 \left( \partial_{xx} u \right) \right] = 0 \\ \frac{U^2}{X} \left[ \frac{X}{\delta} \left( \partial_y P \right) + \frac{\delta}{X} \left( u \partial_x v + v \partial_y v - \partial_{yy} v \right) - \left( \frac{\delta}{X} \right)^3 \partial_{xx} v \right] = 0 \\ \frac{U}{X} \left( \partial_x u + \partial_y v \right) = 0 \ , \end{cases}\end{split}\]

If \(X \gg \delta\) (or equivalently \(U \gg V\)),

in the \(x\)-component of the momentum equation \(\partial_{yy} u\) prevails over \(\partial_{xx} u\); the convective term and the \(x\)-component of the pressure gradient has the same order of magnitude, given \(P \sim \rho U^2\); the leading contribution of the viscous term has the same order of magnitude if \(\frac{\nu X}{U \delta^2} \sim 1\), and thus

\[\delta(x) \sim \left( \frac{\nu x}{U} \right)^{\frac{1}{2}} = x \left( \frac{\nu}{x U} \right)^{\frac{1}{2}} = x \, \text{Re}_x^{-\frac{1}{2}} \ ,\]

i.e.

\[\delta(x) \sim x^{\frac{1}{2}} \ .\]

In order to have thin boundary layers, \(\delta(x) \ll x\), i.e. viscoisty should be «small enough» to be dominated by convection, except for the \(\nu \partial_{yy} u\) contribution in the thin boundary layer region. Once divided by \(\frac{U^2}{X}\) the leading order of the \(x\)-component of the momentum equation is \(\sim 1\).
in the \(y\)-component of the momentum equation, the leading term is the \(y\)-component of the pressure gradient, \(\partial_y P\), with order \(\varepsilon^{-1}\), being \(\varepsilon := \frac{\delta}{X}\)

Retaining the leading components of the equations, Navier-Stokes equation reduces to Prandtl equations, whose non-dimensional form reads

\[\begin{split}\begin{cases} u \partial_x u + v \partial_y u - \partial_{yy} u + \partial_x P = 0 \\ \partial_y P = 0 \\ \partial_x u + \partial_y v = 0 \ , \end{cases}\end{split}\]

todo

Discuss the mathematical nature of Prandtl equations (parabolic equation?): which boundary conditions? Examples for different flows: b.l. on flat surface, jet flows, shear layers, wakes behing a body,…
Pressure across boundary layers is approximately constant, as \(\partial_y P = 0\)

9.2. Integral thicknesses#

9.3. Von Karman integral equation#

9.3.1. Integration of Von Karman integral equation#

9.3.2. Thwaites method#

9.4. Self-similar laminar boundary layer flows#

9.4.1. Boundary layer on flat plate: Blasius solution#

Boundary conditions read

\[\begin{split}\begin{aligned} u(x=0, y ) & = U \\ u(x , y=0 ) & = 0 \\ v(x , y=0 ) & = 0 \\ u(x , y=+\infty) & = U \\ \end{aligned}\end{split}\]

Assuming irrotational flow outside the b.l., pressure is uniform in the whole domain: as the velocity is uniform in the irrotational region, pressure is uniform there, as a consequence of Bernoulli’s theorems. As pressure is constant across the boundary layer in the orthogonal direction, pressure is uniform both inside and outside the boundary layer, \(P(x,y) = \overline{P}\).

Without any reference length scale, a self-similar solution may exist: for a self-similar flow, fields are function of a self-similarity coordindate \(\eta(x,y) = \frac{y}{\delta(x)}\) and not of \(x\), \(y\) independently.

Using stream-function \(\psi\), \(u = \psi_{/y}\), \(v = - \psi_{/x}\), incompressibility constraint is identically satisfied. The last equation to deal with is the \(x\)-component of the momentum equation, that becomes

\[\psi_{/y} \psi_{/yx} - \psi_{/x} \psi_{/yy} - \nu \psi_{/yyy} = 0 \ .\]

Let \(\psi(x,y) = U \delta(x) g\left( \eta(x,y) \right)\), the components of the velocity field becomes

\[\begin{split}\begin{aligned} \frac{u}{U} = \psi_{/y} & = \delta(x) g'(\eta) \eta_{/y} = \\ & = \delta(x) g'(\eta) \frac{1}{\delta(x)} = \\ & = g'(\eta) \\ \frac{v}{U} = - \psi_{/x} & = \delta'(x) g(\eta) + \delta(x) g'(\eta) \eta_{/x} = \\ & = \delta'(x) g(\eta) - \delta(x) g'(\eta) \frac{y \delta'(x)}{\delta^2(x)} = \\ & = \delta'(x) g(\eta) - \eta g'(\eta) \delta'(x) \end{aligned}\end{split}\]

while all the other derivatives appearing in the \(x\)-component of the momentum equation read

\[\begin{split}\begin{aligned} \psi_{/yy} & = g''(\eta) \frac{1}{\delta(x)} \\ \psi_{/yyy} & = g'''(\eta) \frac{1}{\delta^2(x)} \\ \psi_{/yx} & = - g''(\eta) \frac{y \delta'(x)}{\delta^2(x)} = - \eta g''(\eta) \frac{\delta'(x)}{\delta(x)} \\ \end{aligned}\end{split}\]

Inserting these expression in the \(x\)-component of the momentum equation,

\[\begin{split}\begin{aligned} 0 & = - U^2 g' g'' \eta \frac{\delta'}{\delta} - \left( \delta' g - \eta g' \delta' \right) g'' \frac{1}{\delta} - \nu U g''' \frac{1}{\delta^2} = \\ & = - U^2 g g'' \frac{\delta'}{\delta} - \nu U g''' \frac{1}{\delta^2} \ , \end{aligned}\end{split}\]

or

\[0 = \frac{U \delta'(x) \delta(x)}{\nu} g(\eta) g''(\eta) + g'''(\eta) \ .\]

In order to get a self-similar solution, no term depending only on \(x\) or \(y\) may appear in the equation. Thus, the coefficient of the first term must be constant,

\[C = \frac{U \delta'(x) \delta(x)}{\nu} = \frac{U}{\nu} \dfrac{d}{dx} \dfrac{\delta^2(x)}{2} \ ,\]

and usually \(C\) is chosen to be \(C = \frac{1}{2}\) (this choice just changes the definition of the arbitrary thickness of the b.l. \(\delta(x)\) by a multiplicative factor. Solving this equation, the arbitrary thickness of the b.l. goes as

\[\delta(x) - \delta(x_0) = \sqrt{ \frac{\nu}{U} ( x - x_0 ) } \ ,\]

while the \(x\)-component of the momentum equation becomes

\[\frac{1}{2} g g'' + g''' = 0 \ ,\]

supplied with the boundary conditions

\[\begin{split}\begin{aligned} g(0) & = 0 \\ g'(0) & = 0 \\ g'(+\infty) & = 1 \ , \end{aligned}\end{split}\]

for the b.l. on a flat plate.