### 2D BOUNDARY LAYER MODELLING

The height of the boundary layer

**δ**can be estimated as the point where

**u = 0.99 x U**. Here

_{δ}**U**is the velocity of the inviscid flow field outside the viscous boundary layer. In this analysis

_{δ}**U**is obtained from the previous potential flow panel method solution as the tangential surface velocities on the surface of each of the panels.

_{δ} is the pressure gradient along the aerofoil surface (assuming static pressure **P** is constant through the depth of the boundary layer).

is the shear stress on the surface due to friction with the air flow.

for the inviscid external flow where **P**_{o} is the stagnation pressure of the flow.

For incompressible flows the stagnation pressure is constant so that

The governing equations for this simple one-dimensional flow model can be found by applying the laws of conservation of mass and momentum to a control volume covering the boundary layer.

Conservation of momentum :

Mass and Momentum flow will occur at the three free surfaces (1) , (2) and (3) of the control volume.

Thus

The momentum balance will be,

Equating momentum change to applied forces give

**dx**gives,

Displacement effect or lost mass flow,

and Wall shear stress,

These definitions are applied to low speed incompressible flow where **ρ** is constant through the boundary layer. If the above momentum equation is divided by external momentum, and the above profile definitions are substituted in, then the following form is obtained,

**d**x, to find the momentum loss at any point. To simplify calculations, curvature of the aerofoil surface is ignored and assumed polynomial shapes for the boundary layer profile are used to predict local expressions for

**δ, δ**and

^{*}, θ**τ**.

_{w}### Laminar boundary layer.

Initially the flow will be laminar. An exact solution with zero pressure gradient has been predicted by Blassius, so the expected shape for the aerofoil flow will be based on a polynomial fit to this solution, plus an addition term to handle the pressure gradient effect.

where and

**Λ = 7.052**at the stagnation point,

**Λ > 0**for accelerating flow,

**Λ = 0**when there is no pressure gradient (Blassius solution),

**Λ < 0**for decelerating flow and

**Λ = -12**where the boundary layer flow starts to separate.

Boundary layer shapes will then appear as shown in the following figure and will depend on their location along the section and the local pressure gradient (velocity gradient).

**θ**as the primary variable leads to,

**x**to obtain a solution along the boundary layer.

**U**and

_{δ}=0**θ=0**, so the result of the integration becomes,

**x**),

for -0.1<K<0

### Transition

At low angles of attack, transition is usually caused by the growth of natural instabilities at high Reynolds number and by the onset of an adverse pressure gradient.

An empirical prediction of this effect has been obtained by Michel, such that transition occurs when,

**Λ=-12**.

However it is difficult to assess whether a laminar bubble transition occurs or whether the laminar flow completely separates and stall occurs. This prediction is getting very close to the limit of application of this simple theory and care should be taken in its use.

### Turbulent Boundary Layers

Once the surface layer becomes unstable the flow breaks up into random turbulence. The profile at a particular location is no longer fixed but varies with time. In this case profiles can only be analysed as time averages with an associated level of turbulence at each height.

__Instantaneous velocities in turbulent layer at three separate times.__

__Time averaged velocity profile also showing variation of turbulence level through the layer.__

The time average profile is itself rather complex with three distinct regions. At the bottom very close to the surface, is a laminar sublayer which is stable due to the low velocity but has a high shear. At the edge near the invisicid external flow is turbulent decay layer or wake like region and joining these two in the middle is a highly turbulent mixing layer.

For the purposes of our momentum solution we use an approximate polynomial fit to this complex shape,

**H = 9/7**so that a standard 1/7th power law profile is obtained. Using only small H variations gives reasonable estimate of profiles for accelerating, decelerating or near separation (

**H ≈ 2.6**) turbulent boundary layers.

Treating **H** as a constant value and using the following experimental prediction for shear stress in a turbulent layer,

**θ**at

_{te}**x/c = 1**.

Turbulent boundary layer separation is predicted when H reaches or exceeds approximately **2.6**. If separation hasn't occurred then nominally the H value at the trailing edge is **2.0**, so that **δ ^{*}=2θ** at the trailing edge.

### Drag Coefficient

If momentum loss over the whole section is known then drag of the section can be predicted.Change of flow momentum in-to/out-of the control volume is equal to the sum of the applied forces. The control volume is extended downstream where the pressure influence of the aerofoil is negligible so that outlet will have same stream static pressure as the inlet. Thus the only applied force component will be the support force to hold the section in place and thus will be equivalent to the drag (**D**) on the section.

The momentum balance will be,

**τ**and

_{w}=0**H**can be taken as approximately an average of the trailing edge value (

**H=2**), and the far downstream value (

**H=1**), the momentum equation then becomes,

**7/2**power is dependent on our assumptions of the final shape of the turbulent boundary layer at the trailing edge and the reader may wish to investigate more detailed options to improve the accuracy of the final drag coefficient prediction. The final drag coefficient prediction for the aerofoil section will be