5.4 Blasius solution

Blasius Boundary Layer

The Blasius solution gives an exact similarity solution for the laminar boundary layer on a flat plate at zero incidence. It reduces the Prandtl boundary layer equations to a single third-order ODE, yielding closed-form expressions (in terms of a numerically obtained function) for the velocity profile, boundary layer thicknesses, and skin friction. Nearly every quantitative result you'll encounter in introductory boundary layer analysis traces back to this solution.

Laminar Flow Over a Flat Plate

Consider steady, incompressible, laminar flow approaching a semi-infinite flat plate aligned with the freestream. The incoming velocity $U_\infty$ is uniform and parallel to the surface. At the plate surface the no-slip condition forces the fluid velocity to zero, and viscous diffusion creates a thin region of retarded flow: the boundary layer.

Two features set the stage for the Blasius analysis:

• There is zero pressure gradient along the plate ($dp/dx = 0$), because the outer inviscid flow is uniform.
• The boundary layer thickness $\delta(x)$ grows with distance $x$ from the leading edge, but remains much smaller than $x$ itself, so the thin-layer approximation holds.

Prandtl Boundary Layer Equations

Starting from the full Navier-Stokes equations, Prandtl's order-of-magnitude analysis ($\delta \ll x$) discards streamwise viscous diffusion and the normal-direction momentum equation. For a zero-pressure-gradient flat plate the result is:

Continuity:

$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$

Streamwise momentum:

$u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} = \nu \frac{\partial^2 u}{\partial y^2}$

where $u$ and $v$ are the streamwise and wall-normal velocity components, and $\nu$ is the kinematic viscosity. These two equations, with appropriate boundary conditions, are the starting point for the Blasius solution.

Similarity Solution Approach

The key physical observation is that the velocity profile at every station $x$ has the same shape when plotted against a suitably scaled wall-normal coordinate. This self-similarity means a single independent variable can replace both $x$ and $y$.

To exploit this:

1. Introduce a stream function $\psi(x, y)$ that automatically satisfies continuity: $u = \partial\psi/\partial y$, $v = -\partial\psi/\partial x$.
2. Assume $\psi$ takes the form $\psi = \sqrt{\nu x U_\infty}\; f(\eta)$, where $f$ is a dimensionless function of the similarity variable $\eta$.
3. Substitute into the momentum equation. All $x$-dependence cancels, leaving an ODE for $f(\eta)$.

This reduction from a PDE in two variables to an ODE in one variable is what makes the Blasius problem tractable.

Blasius Similarity Variable

The similarity variable is defined as:

$\eta = y \sqrt{\frac{U_\infty}{\nu x}}$

• $y$ is the distance normal to the plate.
• $U_\infty$ is the freestream velocity.
• $\nu$ is the kinematic viscosity.
• $x$ is the streamwise distance from the leading edge.

Physically, $\eta$ measures how far you are from the wall relative to the local boundary layer scale $\sqrt{\nu x / U_\infty}$. Because that scale grows as $\sqrt{x}$, a fixed value of $\eta$ tracks a fixed fraction of the boundary layer thickness at every station.

Blasius Differential Equation

Substituting the similarity form into the momentum equation yields the Blasius equation:

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

with boundary conditions:

• $f(0) = 0$ (no wall-normal velocity at the surface, since $v = 0$ there for an impermeable wall)
• $f'(0) = 0$ (no-slip: $u = 0$ at the wall)
• $f'(\eta \to \infty) = 1$ (velocity matches the freestream far from the wall)

Here primes denote differentiation with respect to $\eta$, and the streamwise velocity is recovered as $u/U_\infty = f'(\eta)$.

Numerical Solution Methods

The Blasius equation is nonlinear and has no closed-form analytical solution. It must be solved numerically.

Shooting method (most common approach):

1. Rewrite the third-order ODE as a system of three first-order ODEs for $f$, $f'$, and $f''$.
2. You know $f(0) = 0$ and $f'(0) = 0$, but $f''(0)$ is unknown. Guess a value for $f''(0)$.
3. Integrate forward using a standard ODE solver (e.g., Runge-Kutta).
4. Check whether $f'(\eta)$ approaches 1 as $\eta \to \infty$ (in practice, at some large $\eta$, say 8–10).
5. Adjust the guess for $f''(0)$ and repeat until the far-field condition is satisfied.

The converged result gives:

$f''(0) \approx 0.3321$

This single number determines the wall shear stress and skin friction for the entire plate.

Finite difference methods offer an alternative: discretize the $\eta$ domain, apply the boundary conditions at both ends, and solve the resulting nonlinear algebraic system iteratively (e.g., Newton's method).

Blasius Velocity Profile

The solution $f'(\eta)$ gives the dimensionless velocity profile $u/U_\infty$ as a function of $\eta$. Its main features:

• At the wall ($\eta = 0$): $u/U_\infty = 0$ (no-slip).
• The profile rises smoothly and monotonically.
• By about $\eta \approx 5.0$, the velocity has reached 99% of $U_\infty$.
• The profile has an inflection-free shape, characteristic of a zero-pressure-gradient boundary layer.

Because of self-similarity, this single curve describes the velocity profile at every streamwise station. The only thing that changes with $x$ is the physical scale of $\eta$.

Boundary Layer Thickness

The 99% boundary layer thickness $\delta$ is the distance from the wall where $u = 0.99\, U_\infty$. From the numerical solution, this occurs at $\eta \approx 5.0$, so:

$\delta = 5.0 \sqrt{\frac{\nu x}{U_\infty}} = \frac{5.0\, x}{\sqrt{Re_x}}$

where $Re_x = U_\infty x / \nu$ is the local Reynolds number.

Notice that $\delta \propto \sqrt{x}$: the boundary layer grows parabolically along the plate. Doubling the distance from the leading edge increases the thickness by a factor of $\sqrt{2} \approx 1.41$, not by a factor of 2.

Displacement Thickness

The displacement thickness $\delta^*$ quantifies how far the outer streamlines are pushed away from the wall by the slow-moving fluid in the boundary layer:

$\delta^* = \int_0^\infty \left(1 - \frac{u}{U_\infty}\right) dy$

For the Blasius profile this evaluates to:

$\delta^* = \frac{1.721\, x}{\sqrt{Re_x}}$

You can think of $\delta^*$ as the thickness of a zero-velocity layer that would produce the same mass flow deficit as the actual boundary layer. It's the relevant thickness when computing how the boundary layer modifies the effective body shape seen by the outer inviscid flow.

Momentum Thickness

The momentum thickness $\theta$ measures the loss of streamwise momentum due to the boundary layer:

$\theta = \int_0^\infty \frac{u}{U_\infty}\left(1 - \frac{u}{U_\infty}\right) dy$

For the Blasius solution:

$\theta = \frac{0.664\, x}{\sqrt{Re_x}}$

Momentum thickness connects directly to drag through the von Kármán integral momentum equation. For a zero-pressure-gradient flat plate, the total drag on a plate of length $L$ (per unit span) equals $\rho U_\infty^2\, \theta(L)$.

The ratio $H = \delta^*/\theta$ is the shape factor. For the Blasius profile, $H \approx 2.59$. This value is a useful reference: shape factors significantly above 2.59 indicate an adverse-pressure-gradient boundary layer approaching separation.

Wall Shear Stress

The wall shear stress is the viscous force per unit area exerted on the plate:

$\tau_w = \mu \left(\frac{\partial u}{\partial y}\right)_{y=0}$

Using the similarity transformation and $f''(0) = 0.3321$:

$\tau_w = 0.3321\, \rho U_\infty^2 \frac{1}{\sqrt{Re_x}}$

The wall shear stress decreases as $x^{-1/2}$: it's highest near the leading edge (where the boundary layer is thinnest and the velocity gradient steepest) and diminishes downstream.

Skin Friction Coefficient

The local skin friction coefficient non-dimensionalizes the wall shear stress:

$C_f = \frac{\tau_w}{\frac{1}{2}\rho U_\infty^2} = \frac{0.664}{\sqrt{Re_x}}$

This is one of the most-cited results from the Blasius solution. For a plate of length $L$, the average (total) skin friction coefficient is obtained by integrating over the plate:

$\bar{C}_f = \frac{1.328}{\sqrt{Re_L}}$

where $Re_L = U_\infty L / \nu$. Note that $\bar{C}_f = 2\, C_f(L)$, a consequence of the $x^{-1/2}$ dependence.

Drag Force on a Flat Plate

For a laminar boundary layer at zero pressure gradient, the drag is entirely due to skin friction (no pressure drag on an aligned flat plate). The drag force per unit span on a plate of length $L$ is:

$D = \int_0^L \tau_w\, dx = \frac{1}{2}\rho U_\infty^2 L\, \bar{C}_f = \frac{0.664\, \rho U_\infty^2 L}{\sqrt{Re_L}}$

This can equivalently be written as $D = \rho U_\infty^2\, \theta(L)$, confirming the connection to momentum thickness.

Blasius Solution Assumptions

The Blasius solution rests on several specific assumptions. Violating any of them means the solution no longer applies directly:

• Steady flow (no time dependence)
• Incompressible fluid (constant density; valid for Mach numbers below about 0.3)
• Laminar flow throughout the boundary layer
• Zero pressure gradient (uniform freestream velocity)
• No-slip condition at an impermeable wall
• Negligible streamwise viscous diffusion (the standard boundary layer approximation, valid when $Re_x \gg 1$)

Validity and Limitations

The Blasius solution accurately describes the boundary layer for:

• Moderate local Reynolds numbers, roughly $Re_x \lesssim 5 \times 10^5$
• Regions not too close to the leading edge (the boundary layer approximation breaks down as $x \to 0$ where $\delta$ is not thin relative to $x$)

It breaks down when:

• The flow transitions to turbulence (see below)
• A pressure gradient is present (use the Falkner-Skan equation for wedge flows, or numerical methods for arbitrary pressure gradients)
• Compressibility effects become significant
• Suction or blowing is applied at the wall

Despite these limitations, the Blasius solution serves as the benchmark against which more complex boundary layer solutions and turbulence models are compared.

Transition to Turbulence

The laminar boundary layer described by the Blasius solution becomes unstable at sufficiently high Reynolds numbers. The critical value is commonly quoted as:

$Re_{x,\text{crit}} \approx 5 \times 10^5$

though in practice this depends on:

• Freestream turbulence intensity (higher turbulence triggers earlier transition)
• Surface roughness (rough surfaces promote transition)
• Pressure gradient (adverse gradients destabilize; favorable gradients delay transition)

The transition process involves growth of Tollmien-Schlichting instability waves, secondary instabilities, and eventual breakdown into fully turbulent flow. The transition region itself is neither purely laminar nor fully turbulent.

Comparison with Turbulent Boundary Layers

Understanding how the Blasius (laminar) solution differs from turbulent boundary layers helps you appreciate when each model applies:

The fuller turbulent profile means more momentum near the wall, which makes turbulent boundary layers more resistant to separation but also produces higher skin friction. This trade-off is central to many aerodynamic design decisions.