# Blasius boundary layer

In physics and fluid mechanics, a Blasius boundary layer (named after Paul Richard Heinrich Blasius) describes the steady two-dimensional laminar boundary layer that forms on a semi-infinite plate which is held parallel to a constant unidirectional flow. Falkner and Skan later generalized Blasius' solution to wedge flow, i.e. flows in which the plate is not parallel to the flow.

## Prandtl's boundary layer equations

A schematic diagram of the Blasius flow profile. The streamwise velocity component ${\displaystyle u(\eta )/U(x)}$ is shown, as a function of the similarity variable ${\displaystyle \eta }$.

Using scaling arguments, Ludwig Prandtl[1] has argued that about half of the terms in the Navier-Stokes equations are negligible in boundary layer flows (except in a small region near the leading edge of the plate). This leads to a reduced set of equations knows as the boundary layer equations. For steady incompressible flow with constant viscosity and density, these read:

Continuity: ${\displaystyle {\dfrac {\partial u}{\partial x}}+{\dfrac {\partial v}{\partial y}}=0}$

${\displaystyle x}$-Momentum: ${\displaystyle u{\dfrac {\partial u}{\partial x}}+v{\dfrac {\partial u}{\partial y}}=-{\dfrac {1}{\rho }}{\dfrac {\partial p}{\partial x}}+{\nu }{\dfrac {\partial ^{2}u}{\partial y^{2}}}}$

${\displaystyle y}$-Momentum: ${\displaystyle {\dfrac {\partial p}{\partial y}}=0}$

Here the coordinate system is chosen with ${\displaystyle x}$ pointing parallel to the plate in the direction of the flow and the ${\displaystyle y}$ coordinate pointing towards the free stream, ${\displaystyle u}$ and ${\displaystyle v}$ are the ${\displaystyle x}$ and ${\displaystyle y}$ velocity components, ${\displaystyle p}$ is the pressure, ${\displaystyle \rho }$ is the density and ${\displaystyle \nu }$ is the kinematic viscosity.

These three partial differential equations for ${\displaystyle u}$, ${\displaystyle v}$ and ${\displaystyle p}$ can be reduced to a single equation for ${\displaystyle u}$ as follows

• By integrating the continuity equation over ${\displaystyle y}$, ${\displaystyle v}$ can be expressed as a function of ${\displaystyle u}$:

${\displaystyle v=-\int {\dfrac {\partial u}{\partial x}}dy}$

• The ${\displaystyle y}$-momentum equation implies that the pressure in the boundary layer must be equal to that of the free stream for any given ${\displaystyle x}$ coordinate. Because the velocity profile is flat in the free stream, there are no viscous effects and Bernoulli's law applies:

${\displaystyle {\dfrac {p}{\rho }}+{\dfrac {U_{e}^{2}}{2}}=}$ constant or, after differentiation: ${\displaystyle {\dfrac {1}{\rho }}{\dfrac {dp}{dx}}=-U_{e}{\dfrac {dU_{e}}{dx}}}$ Here ${\displaystyle U_{e}(x)}$ is the velocity of the fluid outside the boundary layer and is solution of Euler equations (fluid dynamics). The derivatives are not partials because there is no variation with respect to the ${\displaystyle y}$ coordinate.

Substitution of these results into the ${\displaystyle x}$-momentum equations gives:

${\displaystyle u{\dfrac {\partial u}{\partial x}}-{\dfrac {\partial u}{\partial y}}\int {\dfrac {\partial u}{\partial x}}dy=U_{e}{\dfrac {dU_{e}}{dx}}+{\nu }{\dfrac {\partial ^{2}u}{\partial y^{2}}}}$

A number of similarity solutions to this equation have been found for various types of flow, including flat plate boundary layers. The term similarity refers to the property that the velocity profiles at different positions in the flow are the same apart from a scaling factor. These solutions are often presented in the form of non-linear ordinary differential equations.

## Blasius equation - First-order boundary layer

Blasius[2] proposed a similarity solution for the case in which the free stream velocity is constant, ${\displaystyle U_{e}=U,dU_{e}/dx=0}$, which corresponds to the boundary layer over a flat plate that is oriented parallel to the free flow. First he introduced the similarity variable

Developing Blasius boundary layer (not to scale). The velocity profile ${\displaystyle f'}$ is shown in red at selected positions along the plate. The blue lines represent, in top to bottom order, the 99% free stream velocity line (${\displaystyle \delta _{99\%},\eta \approx 3.5}$), the displacement thickness (${\displaystyle \delta _{*},\eta \approx 1.21}$) and ${\displaystyle \delta (x)}$ (${\displaystyle \eta =1}$). See Boundary layer thickness for a more detailed explanation.
${\displaystyle \eta ={\dfrac {y}{\delta (x)}}=y{\sqrt {\dfrac {U}{\nu x}}}}$

Where ${\displaystyle \delta (x)={\sqrt {\nu x/U}}}$ is proportional to the boundary layer thickness. . Next Blasius proposed the stream function

${\displaystyle \psi ={\sqrt {\nu Ux}}f(\eta )}$

in which the newly introduced normalized stream function, ${\displaystyle f(\eta )}$, is only a function of the similarity variable. This leads directly to the velocity components

${\displaystyle u(x,y)={\dfrac {\partial \psi }{\partial y}}=Uf'(\eta )}$
${\displaystyle v(x,y)=-{\dfrac {\partial \psi }{\partial x}}={\sqrt {\dfrac {\nu U}{x}}}(\eta f'-f)}$

Where the prime denotes derivation with respect to ${\displaystyle \eta }$. Substitution into the momentum equation gives the Blasius equation

${\displaystyle f'''+{\frac {1}{2}}f''f=0}$

The boundary conditions are the no-slip condition

${\displaystyle u(x,0)=0\rightarrow f'(0)=0}$

impermeability of the wall

${\displaystyle v(x,0)=0\rightarrow f(0)=0}$

and the free stream velocity outside the boundary layer

${\displaystyle u(x,\infty )=U\rightarrow f'(\infty )=1}$

This is a third order non-linear ordinary differential equation which can be solved numerically, e.g. with the shooting method. The limiting form for small ${\displaystyle \eta <<1}$ is

${\displaystyle f(\eta )={\frac {1}{2}}\alpha \eta ^{2}+O(\eta ^{5}),\qquad \alpha =0.332}$

and the limiting form for large ${\displaystyle \eta >>1}$ is

${\displaystyle f(\eta )=\eta -\beta +\quad \mathrm {exponentially} \quad \mathrm {small} \quad \mathrm {terms} ,\qquad \beta =1.72}$

The appropriate parameters to compare with the experimental observations are displacement thickness ${\displaystyle \delta ^{*}}$, momentum thickness ${\displaystyle \theta }$ wall shear stress ${\displaystyle \tau _{w}}$ and drag force ${\displaystyle F}$ acting on a length ${\displaystyle l}$ of the plate, which are given for the Blasius profile

${\displaystyle \delta ^{*}=\int _{0}^{\infty }\left(1-{\frac {u}{U}}\right)dy=1.72{\sqrt {\frac {\nu x}{U}}}}$
${\displaystyle \theta =\int _{0}^{\infty }{\frac {u}{U}}\left(1-{\frac {u}{U}}\right)dy=0.665{\sqrt {\frac {\nu x}{U}}}}$
${\displaystyle \tau _{w}=\mu \left({\frac {\partial u}{\partial y}}\right)_{y=0}=0.332{\sqrt {\frac {\rho \mu U^{3}}{x}}}}$
${\displaystyle F=2\int _{0}^{\infty }\tau _{w}dx=1.328{\sqrt {\rho \mu lU^{3}}}}$

The factor ${\displaystyle 2}$ in the drag force formula is to account both sides of the plate.

### Uniqueness of Blasius solution[3]

The Blasius solution is not unique from a mathematical perspective as Ludwig Prandtl himself noted it and analyzed by series of researchers such as Keith Stewartson, Paul A. Libby etc. To this solution, any one of the infinite discrete set of eigenfunctions can be added, each of which satisfies the linearly perturbed equation with homogeneous conditions and exponential decay at infinity. The first of these eigenfunctions turns out be the ${\displaystyle x}$ derivative of the first order Blasius solution, which represents the uncertainty in the effective location of the origin.

### Second-order boundary layer

This boundary layer approximation predicts a non-zero vertical velocity far away from the wall, which needs to be accounted in next order outer inviscid layer and the corresponding inner boundary layer solution, which in turn will a predict a new vertical velocity and so on. The vertical velocity at infinity for the first order boundary layer problem from the Blasius equation is

${\displaystyle v=0.86{\sqrt {\frac {\nu U}{x}}}}$

## Blasius boundary layer with suction on the wall

Suction is one of the common methods employed to postpond the boundary layer separation. Consider a uniform suction velocity at the wall ${\displaystyle v(0)=-V}$. For distances from the leading edge of the plate ${\displaystyle x>>\nu U/V^{2}}$, both the boundary layer thickness and the solution are independent of ${\displaystyle x}$ given by

${\displaystyle \delta ={\frac {\nu }{V}},\quad u=U(1-e^{-yV/\nu }),\quad v=-V}$

For distances ${\displaystyle x\sim \nu U/V^{2}}$, there is no self-similar solution, the whole boundary layer equations need to be integrated numerically.

## Falkner–Skan equation

Wedge flow.

We can generalize the Blasius boundary layer by considering a wedge at an angle of attack ${\displaystyle \pi \beta /2}$ from some uniform velocity field ${\displaystyle U_{0}}$. We then estimate the outer flow to be of the form:

${\displaystyle u_{e}(x)=U_{0}\left({\frac {x}{L}}\right)^{m}}$

Where ${\displaystyle L}$ is a characteristic length and m is a dimensionless constant. In the Blasius solution, m = 0 corresponding to an angle of attack of zero radians. Thus we can write:

${\displaystyle \beta ={\frac {2m}{m+1}}.}$

As in the Blasius solution, we use a similarity variable ${\displaystyle \eta }$ to solve the boundary layer equations.

Falkner-Skan boundary layer profiles for selected values of ${\displaystyle m}$.
${\displaystyle \eta =y{\sqrt {\frac {U_{0}(m+1)}{2\nu L}}}\left({\frac {x}{L}}\right)^{(m-1)/2}}$

It becomes easier to describe this in terms of its stream function which we write as

${\displaystyle \psi =U(x)\delta (x)f(\eta )={\sqrt {\frac {2\nu U_{0}L}{m+1}}}\left({\frac {x}{L}}\right)^{(m+1)/2}f(\eta )}$

Thus the initial differential equation which was written as follows:

${\displaystyle u{\partial u \over \partial x}+v{\partial u \over \partial y}=c^{2}mx^{2m-1}+{\nu }{\partial ^{2}u \over \partial y^{2}}.}$

Can now be expressed in terms of the non-linear ODE known as the Falkner–Skan equation (named after V. M. Falkner and Sylvia W. Skan[4]).

${\displaystyle f'''+ff''+\beta \left[1-(f')^{2}\right]=0}$

Here, m < 0 corresponds to an adverse pressure gradient (often resulting in boundary layer separation) while m > 0 represents a favorable pressure gradient. (Note that m = 0 recovers the Blasius equation). In 1937 Douglas Hartree showed that physical solutions to the Falkner–Skan equation exist only in the range -0.0905 ≤ m ≤ 2. For more negative values of m, that is, for stronger adverse pressure gradients, all solutions satisfying the boundary conditions at η = 0 have the property that f(η) > 1 for a range of values of η. This is physically unacceptable because it implies that the velocity in the boundary layer is greater than in the main flow.[5]

Further details may be found in Wilcox (2007).