Boundary layer thickness

The bounded boundary layer description

Bounded boundary layers is a name used to designate fluid flow along an interior wall such that the other interior walls induce a pressure effect on the fluid flow along the wall under consideration. The defining characteristic of this type of boundary layer is that the velocity profile normal to the wall often smoothly asymptotes to a constant velocity value denoted as u_e(x). The bounded boundary layer concept is depicted for steady flow entering the lower half of a thin flat plate 2-D channel of height H in Figure 1 (the flow and the plate extends in the positive/negative direction perpendicular to the x-y-plane). Examples of this type of boundary layer flow occur for fluid flow through most pipes, channels, and wind tunnels. The 2-D channel depicted in Figure 1 is stationary with fluid flowing along the interior wall with time-averaged velocity u(x,y) where x is the flow direction and y is the normal to the wall. The H/2 dashed line is added to acknowledge that this is an interior pipe or channel flow situation and that there is a top wall located above the pictured lower wall. Figure 1 depicts flow behavior for H values that are larger than the maximum boundary layer thickness but less than thickness at which the flow starts to behave as an exterior flow. If the wall-to-wall distance, H, is less than the viscous boundary layer thickness then the velocity profile, defined as u(x,y) at x for all y, takes on a parabolic profile in the y-direction and the boundary layer thickness is just H/2.

At the solid walls of the plate the fluid has zero velocity (no-slip boundary condition), but as you move away from the wall, the velocity of the flow increases without peaking, and then approaches a constant mean velocity u_e(x). This asymptotic velocity may or may not change along the wall depending on the wall geometry. The point where the velocity profile essentially reaches the asymptotic velocity is the boundary layer thickness. The boundary layer thickness is depicted as the curved dashed line originating at the channel entrance in Figure 1. It is impossible to define an exact location at which the velocity profile reaches the asymptotic velocity. As a result, a number of boundary layer thickness parameters, generally denoted as ${\displaystyle \delta (x)}$, are used to describe characteristic thickness scales in the boundary layer region. Also of interest is the velocity profile shape which is useful in differentiating laminar from turbulent boundary layer flows. The profile shape refers to the y-behavior of the velocity profile as it transitions to u_e(x).

The 99% boundary layer thickness

The boundary layer thickness, ${\displaystyle \delta }$, is the distance normal to the wall to a point where the flow velocity has essentially reached the 'asymptotic' velocity, ${\displaystyle u_e}$. Prior to the development of the Moment Method, the lack of an obvious method of defining the boundary layer thickness led much of the flow community in the later half of the 1900s to adopt the location ${\displaystyle y_{99}}$, denoted as ${\displaystyle {\delta }_{99}}$ and given by

${\displaystyle u(x,y_{99})=0.99u_e(x),}$

as the boundary layer thickness.

For laminar boundary layer flows along a flat plate channel that behave according to the Blasius solution conditions, the ${\displaystyle {\delta }_{99}}$ value is closely approximated by

${\displaystyle {\delta }_{99}(x)\approx 5.0\sqrt{\frac{\nu x}{u_0}}=5.0\frac{x}{\sqrt{{\mathrm{R}\mathrm{e}}_x}},}$

where ${\displaystyle u_e\approx u_0}$ is constant, and where

${\displaystyle {\mathrm{R}\mathrm{e}}_x}$ is the Reynolds number,

${\displaystyle u_0}$ is the freestream velocity,

${\displaystyle u_e}$ is the asymptotic velocity,

${\displaystyle x}$ is the distance downstream from the start of the boundary layer, and

${\displaystyle \nu }$ is the kinematic viscosity.

For turbulent boundary layers along a flat plate channel, the boundary layer thickness, ${\displaystyle \delta }$, is given by

${\displaystyle \delta (x)\approx 0.37\frac{x}{{{\mathrm{R}\mathrm{e}}_x}^{1/5}}.}$

This turbulent boundary layer thickness formula assumes 1) the flow is turbulent right from the start of the boundary layer and 2) the turbulent boundary layer behaves in a geometrically similar manner (i.e. the velocity profiles are geometrically similar along with the flow in the x-direction, differing only by scaling parameters in ${\displaystyle y}$ and ${\displaystyle u(x,y)}$). Neither one of these assumptions is true for the general turbulent boundary layer case so care must be exercised in applying this formula.

Displacement thickness

The displacement thickness, ${\displaystyle {\delta }_1}$ or ${\displaystyle {\delta }^{\ast }}$, is the normal distance to a reference plane representing the lower edge of a hypothetical inviscid fluid of uniform velocity ${\displaystyle u_e}$ that has the same flow rate as occurs in the real fluid with the boundary layer.

The displacement thickness essentially modifies the shape of a body immersed in a fluid to allow, in principle, an inviscid solution if the displacement thicknesses were known a priori.

The definition of the displacement thickness for compressible flow, based on mass flow rate, is

${\displaystyle {\delta }_1(x)={\int }_0^{H/2}\left(1-\frac{\rho (x,y)u(x,y)}{{\rho }_e{u}_e(x)}\right)\mathrm{d}y,}$

where ${\displaystyle \rho (x,y)}$ is the density. For incompressible flow, the density is constant so the definition based on volumetric flow rate becomes

${\displaystyle {\delta }_1(x)={\int }_0^{H/2}\left(1-\frac{u(x,y)}{u_e(x)}\right)\mathrm{d}y.}$

For turbulent boundary layer calculations, the time-averaged density and velocity are used.

For laminar boundary layer flows along a flat plate that behave according to the Blasius solution conditions, the displacement thickness is

${\displaystyle {\delta }_1(x)\approx 1.72\sqrt{\frac{\nu x}{u_0}},}$

where ${\displaystyle u_e\approx u_0}$ is constant.

The displacement thickness is not directly related to the boundary layer thickness but is given approximately as ${\displaystyle {\delta }_1\approx \delta /3}$. It has a prominent role in calculating the Shape Factor. It also shows up in various formulas in the Moment Method.

Momentum thickness

The momentum thickness, ${\displaystyle \theta }$ or ${\displaystyle {\delta }_2}$, is the normal distance to a reference plane representing the lower edge of a hypothetical inviscid fluid of uniform velocity ${\displaystyle u_e}$ that has the same momentum flow rate as occurs in the real fluid with the boundary layer.

The momentum thickness definition for compressible flow based on the mass flow rate is

${\displaystyle {\delta }_2(x)={\int }_0^{H/2}\frac{\rho (x,y)u(x,y)}{{\rho }_e{u}_e(x)}\left(1-\frac{u(x,y)}{u_e(x)}\right)\mathrm{d}y.}$

For incompressible flow, the density is constant so that the definition based on volumetric flow rate becomes

${\displaystyle {\delta }_2(x)={\int }_0^{H/2}\frac{u(x,y)}{u_e(x)}\left(1-\frac{u(x,y)}{u_e(x)}\right)\mathrm{d}y,}$

where ${\displaystyle \rho }$ are the density and ${\displaystyle u_e}$ is the 'asymptotic' velocity.

For turbulent boundary layer calculations, the time averaged density and velocity are used.

For laminar boundary layer flows along a flat plate that behave according to the Blasius solution conditions, the momentum thickness is

${\displaystyle {\delta }_2(x)\approx 0.664\sqrt{\frac{\nu x}{u_0}},}$

where ${\displaystyle u_e\approx u_0}$ is constant.

The momentum thickness is not directly related to the boundary layer thickness but is given approximately as ${\displaystyle {\delta }_2\approx \delta /6}$. It has a prominent role in calculating the Shape Factor.

A related parameter called the Energy Thickness is sometimes mentioned in reference to turbulent energy distribution but is rarely used.

Shape factor

A shape factor is used in boundary layer flow to help to differentiate laminar and turbulent flow. It also shows up in various approximate treatments of the boundary layer including the Thwaites method for laminar flows. The formal definition is given by

${\displaystyle H_{12}(x)=\frac{{\delta }_1(x)}{{\delta }_2(x)},}$

where ${\displaystyle H_{12}}$ is the shape factor, ${\displaystyle {\delta }_1}$ is the displacement thickness and ${\displaystyle {\delta }_2}$ is the momentum thickness.

Conventionally, ${\displaystyle H_{12}}$ = 2.59 (Blasius boundary layer) is typical of laminar flows, while ${\displaystyle H_{12}}$ = 1.3 - 1.4 is typical of turbulent flows near the laminar-turbulent transition. For turbulent flows near separation, ${\displaystyle H_{12}\approx }$2.7. The dividing line defining laminar-transitional and transitional-turbulent ${\displaystyle H_{12}}$ values is dependent on a number of factors so it is not always a definitive parameter for differentiating laminar, transitional, or turbulent boundary layers.

Moment method

A relatively new method for describing the thickness and shape of the boundary layer uses the mathematical moment methodology which is commonly used to characterize statistical probability functions. The boundary layer moment method was developed from the observation that the plot of the second derivative of the Blasius boundary layer for laminar flow over a plate looks very much like a Gaussian distribution curve. The implication of the second derivative Gaussian-like shape is that the velocity profile shape for laminar flow is closely approximated as a twice integrated Gaussian function.

The moment method is based on simple integrals of the velocity profile that use the entire profile, not just a few tail region data points as does ${\displaystyle {\delta }_{99}}$. The moment method introduces four new parameters that help describe the thickness and shape of the boundary layer. These four parameters are the mean location, the boundary layer width, the velocity profile skewness, and the velocity profile excess. The skewness and excess are true shape parameters as opposed to the simple ratio parameters like the H₁₂. Applying the moment method to the first and second derivatives of the velocity profile generates additional parameters that, for example, determine the location, shape, and thickness of the viscous forces in a turbulent boundary layer. A unique property of the moment method parameters is that it is possible to prove that many of these velocity thickness parameters are also similarity scaling parameters. That is, if similarity is present in a set of velocity profiles, then these thickness parameters must also be similarity length scaling parameters.

It is straightforward to cast the properly scaled velocity profile and its first two derivatives into suitable integral kernels.

The central moments based on the scaled velocity profiles are defined as

${\displaystyle {\zeta }_n(x)={\int }_0^{H/2}(y-m(x){)}^n\frac{1}{{\delta }_1(x)}\left(1-\frac{u(x,y)}{u_e(x)}\right)\mathrm{d}y,}$

where ${\displaystyle {\delta }_1(x)}$ is the displacement thickness and the mean location, ${\displaystyle m(x)}$ is given by

${\displaystyle m(x)={\int }_0^{H/2}y\frac{1}{{\delta }_1(x)}\left(1-\frac{u(x,y)}{u_e(x)}\right)\mathrm{d}y.}$

There are some advantages to also include descriptions of moments of the boundary layer profile derivatives with respect to the height above the wall. Consider the first derivative velocity profile central moments given by

${\displaystyle {\kappa }_n(x)={\int }_0^{H/2}(y-{\delta }_1(x){)}^n\frac{d\{u(x,y)/u_e(x)\}}{dy}\mathrm{d}y,}$

where the first derivative mean location is the displacement thickness ${\displaystyle {\delta }_1(x)}$.

Finally the second derivative velocity profile central moments are given by

${\displaystyle {\lambda }_n(x)={\int }_0^{H/2}(y-{\mu }_1(x){)}^n\frac{d^2\{-{\mu }_1(x)u(x,y)/u_e(x)\}}{d{y}^2}\mathrm{d}y,}$

where the second derivative mean location, ${\displaystyle {\mu }_1(x)}$, is given by

${\displaystyle {\mu }_1(x)=\frac{u_e(x)}{{\frac{du(x,y)}{dy}|}_{y=0}}=\frac{\upsilon u_e(x)}{{\tau }_w(x)},}$

where ${\displaystyle \upsilon }$ is the viscosity and where ${\displaystyle {\tau }_w(x)}$ is the wall shear stress. The mean location, ${\displaystyle {\mu }_1}$, for this case is formally defined as u_e(x) divided by the area under the second derivative curve.

The above equations work for both laminar and turbulent boundary layers as long as the time-averaged velocity is used for the turbulent case.

With the moments and the mean locations defined, the boundary layer thickness and shape can be described in terms of the boundary layer widths (variance), skewnesses, and excesses (excess kurtosis). Experimentally, it is found that the thickness defined as ${\displaystyle {\delta }_m=m+3{\sigma }_m}$ where ${\displaystyle {\sigma }_m={\zeta }_2^{1/2}}$, tracks the ${\displaystyle {\delta }_{99}}$ very well for turbulent boundary layer flows.

Taking a cue from the boundary layer momentum balance equations, the second derivative boundary layer moments, ${\displaystyle {\lambda }_n}$ track the thickness and shape of that portion of the boundary layer where the viscous forces are significant. Hence the moment method makes it possible to track and quantify the laminar boundary layer and the inner viscous region of turbulent boundary layers using ${\displaystyle {\lambda }_n}$ moments whereas the boundary layer thickness and shape of the total turbulent boundary layer is tracked using ${\displaystyle {\zeta }_n}$ and ${\displaystyle {\kappa }_n}$ moments.

Calculation of the 2nd derivative moments can be problematic since under certain conditions the second derivatives can become positive in the very near-wall region (in general, it is negative). This appears to be the case for interior flow with an adverse pressure gradient (APG). Integrand values do not change sign in standard probability framework so the application of the moment methodology to the second derivative case will result in biased moment measures. A simple fix is to exclude the problematic values and define a new set of moments for a truncated second derivative profile starting at the second derivative minimum. If the width, ${\displaystyle {\sigma }_v}$, is calculated using the minimum as the mean location, then the viscous boundary layer thickness, defined as the point where the second derivative profile becomes negligible above the wall, can be properly identified with this modified approach.

For derivative moments whose integrands do not change sign, the moments can be calculated without the need to take derivatives by using integration by parts to reduce the moments to simply integrals based on the displacement thickness kernel given by

${\displaystyle {\alpha }_n(x)={\int }_0^{H/2}{y}^n\left(1-\frac{u(x,y)}{u_e(x)}\right)\mathrm{d}y.}$

For example, the second derivative ${\displaystyle {\sigma }_v}$ value is ${\displaystyle {\sigma }_v=\sqrt{{-{\mu }_1}^2+2{\mu }_1{\alpha }_0}}$ and the first derivative skewness, ${\displaystyle {\gamma }_1}$, can be calculated as

${\displaystyle {\gamma }_1(x)={\kappa }_3/{\kappa }_2^{3/2}=(2{\delta }_1^3-6{\delta }_1{\alpha }_1+3{\alpha }_2)/(2{\alpha }_1-{\delta }_1^2{)}^{3/2}.}$

This parameter was shown to track the boundary layer shape changes that accompany the laminar to turbulent boundary layer transition.

Numerical errors encountered in calculating the moments, especially the higher-order moments, are a serious concern. Small experimental or numerical errors can cause the nominally free stream portion of the integrands to blow up. There are certain numerical calculation recommendations that can be followed to mitigate these errors.