Boundary Layer theory

The ratio of displacement thickness to boundary layer thickness for the 1/7 power-law profile is $\delta^*/\delta = 1/8 = 0.125$.

Boundary layer

When a real fluid flows past a solid surface, the fluid layer touching the surface sticks to it (no-slip). Velocity rises from zero at the wall to the free-stream velocity $U$ over a thin region near the surface. This thin region, where viscous shear is important, is the boundary layer. Outside it the flow behaves as an ideal (inviscid) fluid.

Boundary layer thickness ($\delta$)

It is the distance from the wall at which the velocity reaches 99% of the free-stream velocity: $u = 0.99\,U$ at $y = \delta$.

Displacement thickness ($\delta^*$)

It is the distance by which the external streamlines are shifted away from the wall because of the slow-moving fluid in the boundary layer. Equivalently, it is the thickness of a layer of free-stream flow that has the same discharge as the loss of discharge caused by the boundary layer.

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

Ratio for $u/U = (y/\delta)^{1/7}$

$$\begin{aligned} \delta^* &= \int_0^{\delta}\left[1 - \left(\frac y\delta\right)^{1/7}\right]dy = \delta - \frac{\delta}{1 + 1/7}\\ &= \delta - \frac{7}{8}\delta = \frac{\delta}{8} \end{aligned}$$ $$\frac{\delta^*}{\delta} = \frac18 = 0.125$$

Answer: $\delta^*/\delta = 1/8$, so the displacement thickness is 12.5% of the boundary layer thickness.

Outside the boundary layers the core flow is accelerated, because the displacement thickness reduces the effective flow area. Apply continuity to the core, then Bernoulli. The pressure falls by about 55.7 Pa.

Assumptions

Air density $\rho = 1.23\ \text{kg/m}^3$. The flow in the core is frictionless and incompressible, so Bernoulli applies between sections 1 and 2.

  Section 1                  Section 2
  ----------------------     ----------------------
   U0 = 30 m/s uniform  -->    core U2 (faster)
  ----------------------     = boundary layer (d* = 0.9 mm)

Continuity (effective area)

At section 2 the effective flow area is reduced by $\delta^*$ on each wall:

$$A_{eff} = (76 - 2 \times 0.9)^2 = 74.2^2\ \text{mm}^2,\qquad A_1 = 76^2\ \text{mm}^2$$ $$U_0A_1 = U_2A_{eff} \;\Rightarrow\; U_2 = 30\left(\frac{76}{74.2}\right)^2 = 31.47\ \text{m/s}$$

Bernoulli (horizontal duct)

$$\begin{aligned} p_1 - p_2 &= \tfrac12\rho\,(U_2^2 - U_0^2) \\ &= \tfrac12(1.23)(31.47^2 - 30^2) = 55.7\ \text{Pa} \end{aligned}$$

Answer: $p_2 - p_1 = -55.7$ Pa, i.e. the pressure drops by about 55.7 Pa (about 5.7 mm of water) between sections 1 and 2.

Laminar sub-layer

In a turbulent boundary layer, a very thin layer next to the solid surface still has laminar flow, because the wall damps out the eddies there. In this layer the velocity changes almost linearly with $y$ and the shear is purely viscous, $\tau_0 = \mu\,(du/dy)$. Its thickness is

$$\delta' = \frac{11.6\,\nu}{u_*},\qquad u_* = \sqrt{\tau_0/\rho}$$

It is only a small fraction of the total boundary layer thickness, but it controls wall shear and heat transfer, and decides whether a surface is hydraulically smooth or rough.

Laminar vs turbulent boundary layer

When a uniform stream of velocity $U$ flows parallel to a thin flat plate, the fluid at the surface sticks to it (no-slip). Viscous shear slows the layers above, and a region of velocity gradient grows along the plate. This is the boundary layer. Its thickness $\delta$ increases with distance $x$ from the leading edge.

Development

 U ->  free stream (inviscid)
 ---------------------------------------------------
        .---- laminar ----.transition.---- turbulent ---
 U -> ----------___________......_____________________
                   boundary layer   (laminar sub-layer
 =================================   at wall)
 leading edge         x_cr        flat plate -->

1. Laminar zone: Starting at the leading edge, the layer is thin and the flow is orderly, in smooth layers. Thickness grows as $\delta \propto \sqrt{x}$ (Blasius, $\delta = 5x/\sqrt{Re_x}$).
2. Transition zone: At a critical Reynolds number $Re_{x,cr} \approx 5\times10^5$ (for a smooth plate and low free-stream turbulence), disturbances amplify and the flow starts to change to turbulent.
3. Turbulent zone: Beyond transition, mixing by eddies makes the layer thicker ($\delta \propto x^{0.8}$) with a fuller velocity profile. A very thin laminar sub-layer remains next to the wall.

where $Re_x = Ux/\nu$.

Salient features

• Velocity is zero at the wall and reaches $0.99U$ at $y = \delta$.
• $\delta$ increases downstream; the pressure gradient along a flat plate is zero.
• Wall shear stress is highest near the leading edge in the laminar region, drops, then jumps at transition and is higher in the turbulent region than for laminar flow at the same $x$.
• A larger viscosity or lower velocity (lower $Re$) gives a thicker layer.
• Pressure across the layer is nearly constant, $\partial p/\partial y \approx 0$.
• The effect of the plate on the flow outside the boundary layer is represented by the displacement thickness $\delta^*$.

Characteristics of the flow

A flat plate held parallel to a free stream develops a boundary layer, in which viscosity slows the fluid near the surface. Velocity is zero at the wall and equals the free-stream velocity $U$ at the edge of the layer ($y = \delta$). The thickness $\delta$ grows with distance $x$. The flow is laminar near the leading edge, changes through a transition region and becomes turbulent downstream, with the change at about $Re_x = Ux/\nu \approx 5\times10^5$. Outside the layer the flow is practically inviscid.

Sketch

 U ->
 ---------------------------------------------------
                                      Turbulent BL
          Laminar BL    Transition  _________________
 U -> ----------____________......___________________
                                      ..... laminar sub-layer
 ========================================= plate
 leading edge      x ------------------------>
   (i) laminar  (iii) transition  (ii) turbulent
   (iv) laminar sub-layer: thin film at the wall

• (i) Laminar boundary layer: smooth flow, $\delta = 5x/\sqrt{Re_x}$.
• (iii) Transition zone: flow changes from laminar to turbulent, $Re_x \approx 3\times10^5$ to $5\times10^5$ or more.
• (ii) Turbulent boundary layer: eddy mixing, thicker, fuller velocity profile.
• (iv) Laminar sub-layer: very thin viscous layer next to the wall, even under the turbulent layer.

Displacement thickness

It is the distance by which the outer streamlines are displaced away from the plate because of the reduction of flow in the boundary layer:

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

Find the Reynolds number first to decide whether the boundary layer is laminar or turbulent, then use the matching formulas.

Assumptions

Air at about 20 °C: $\rho = 1.2\ \text{kg/m}^3$, $\nu = 1.5\times10^{-5}\ \text{m}^2/\text{s}$. $U = 6.5$ m/s, $L = 2$ m, $B = 1.5$ m, $x = 1.8$ m.

Reynolds numbers

$$Re_x = \frac{Ux}{\nu} = \frac{6.5(1.8)}{1.5\times10^{-5}} = 7.8\times10^5,\qquad Re_L = \frac{6.5(2)}{1.5\times10^{-5}} = 8.67\times10^5$$

Both exceed the critical value $5\times10^5$, so the boundary layer is taken as turbulent (1/7 power law, valid for $Re < 10^7$).

Boundary layer thickness and displacement thickness at x = 1.8 m

$$\begin{aligned} \delta &= \frac{0.37\,x}{Re_x^{0.2}} = \frac{0.37(1.8)}{(7.8\times10^5)^{0.2}} = 0.0442\ \text{m} \\ \delta^* &= \frac{\delta}{8} = 0.00552\ \text{m} = 5.52\ \text{mm} \end{aligned}$$

Shear stress at x = 1.8 m

$$\begin{aligned} C_f &= \frac{0.0576}{Re_x^{0.2}} = 0.003819 \\ \tau_0 &= C_f\cdot\tfrac12\rho U^2 = 0.003819(0.5)(1.2)(6.5)^2 = 0.0968\ \text{N/m}^2 \end{aligned}$$

Drag force on one face

$$\begin{aligned} C_D &= \frac{0.074}{Re_L^{0.2}} = 0.004805 \\ F_D &= C_D\cdot\tfrac12\rho U^2\,(L B) = 0.004805(0.5)(1.2)(6.5)^2(2\times1.5) = 0.365\ \text{N} \end{aligned}$$

If the laminar length at the leading edge is allowed for, $C_D = 0.074/Re_L^{0.2} - 1700/Re_L = 0.002843$ and $F_D = 0.216$ N.

Answer: $\tau_0 \approx 0.097$ N/m², $\delta^* \approx 5.5$ mm, drag on one face $\approx 0.37$ N (about 0.22 N if the initial laminar part is included).

Boundary layer thickness ($\delta$)

It is the distance from the wall, measured normal to it, at which the velocity reaches 99% of the free-stream velocity $U$ ($u = 0.99U$).

Application: It decides the region where viscous effects matter, so it is used to fix the size of the region to be analysed, to design the spacing of probes and the wall clearance in wind tunnels, and to size the thickness of insulation or coatings in flow.

Displacement thickness ($\delta^*$)

It is the distance by which the external flow is shifted away from the wall because of the velocity deficit in the boundary layer:

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

Application: In wind tunnels, diffusers and ducts the walls are placed (or the area is enlarged) by $\delta^*$ to keep the core velocity constant. It also gives the pressure change in the duct entrance, and the real contour of an aerofoil or body for the outside potential flow.

Momentum thickness ($\theta$)

It is the thickness of fluid layer that, at velocity $U$, carries the momentum lost because of the boundary layer:

$$\theta = \int_0^{\delta}\frac uU\left(1 - \frac uU\right)dy$$

Application: It is used in the momentum integral equation to find the wall shear stress and the drag force on a surface, $\tau_0 = \rho U^2\,d\theta/dx$, and $F_D = \rho U^2\theta\,b$ for a plate of width $b$.

Boundary layer separation

When flow moves into a region of rising pressure (an adverse pressure gradient, $dp/dx > 0$, as on the rear of a cylinder or in a diverging channel), the fluid near the wall loses momentum to friction and cannot go further. The velocity gradient at the wall becomes zero, $(du/dy)_{y=0} = 0$. Beyond this point of separation, the flow near the wall reverses, the boundary layer leaves the surface and a wake of eddies forms.

 free stream ->  ___________________
                 \  boundary layer   separated flow
 body surface ====\=======S==========  S = separation point
                    --> <-- back flow    (du/dy = 0 at wall)

Separation causes high pressure (form) drag and, on a wing, loss of lift (stall).

Stagnation point

It is the point on the surface of a body in a flowing fluid where the fluid velocity becomes zero. The streamline dividing the flow splits here, and the pressure reaches its maximum value (stagnation pressure):

$$p_0 = p + \tfrac12\rho V^2$$

   ---->  ---->  ___
   ---->  ---->  /  |  stagnation point
   ---->  ----> (*  |  at the nose, v = 0

Boundary layer concept

When a real fluid flows past a solid body, fluid at the surface sticks to it (no-slip). Velocity increases from zero at the wall to the free-stream velocity $U$ over a thin layer. Viscous shear is large only within this layer, called the boundary layer. Outside it the flow is treated as ideal (potential) flow. This idea (Prandtl, 1904) lets the flow be split into a thin viscous layer and an outer inviscid region.

Boundary layer thickness ($\delta$)

It is the distance from the wall at which the velocity equals 99% of the free-stream velocity: $u = 0.99U$ at $y = \delta$.

Laminar sub-layer

Under a turbulent boundary layer there is a very thin layer next to the wall in which flow stays laminar and the shear is purely viscous: $\tau_0 = \mu\,du/dy$. Its thickness is $\delta' = 11.6\nu/u_*$.

Point of separation

It is the point on the surface where flow moving against an adverse pressure gradient ($dp/dx > 0$) is brought to rest at the wall, so $(du/dy)_{y=0} = 0$. Downstream the flow near the wall reverses and the boundary layer leaves the surface, forming a wake.

 U -> ===========================
            turbulent BL         separation
                      ~~~~~~~~~~~~  S
 solid ========================\===\====
      laminar sub-layer        <-- back flow

Concept of boundary layer

The boundary layer is the thin layer next to a solid surface in which the fluid velocity changes from zero (at the wall, no-slip) to nearly the free-stream value. Viscous effects are confined to this layer; outside it the flow is inviscid.

Growth in a closed conduit (pipe flow)

 inlet            entrance length Le          fully developed
  ---------------------------------------------------------
   U0 ->  boundary layers grow     core meets   parabolic /
   uniform    from wall inward ->  at centre    turbulent profile
  ---------------------------------------------------------

1. At the pipe entrance the velocity is nearly uniform, $U_0$. A boundary layer begins to grow from the wall at the entrance.
2. The layer thickens along the pipe, slowing the fluid near the wall. By continuity the core fluid, outside the layer, accelerates, and the pressure drops faster than in fully developed flow.
3. At a distance called the entrance (inlet) length $L_e$, the layers from the opposite walls meet at the centre. After this the velocity profile no longer changes with $x$, and the flow is fully developed.
4. Entrance length: laminar flow $L_e \approx 0.05\,Re\,D$; turbulent flow $L_e \approx 4.4\,Re^{1/6}D$ (about 25 to 40 diameters).

Three examples of use of the boundary layer concept

1. Calculation of skin friction drag on ships, aircraft wings and plates.
2. Explaining separation and stall on aerofoils, and the design of streamlined bodies to reduce form drag.
3. Design of wind tunnels, diffusers and ducts (allowing for displacement thickness), and prediction of the heat transfer from surfaces.

Concept of boundary layer

The boundary layer is the thin region next to a solid surface where the velocity changes from zero at the surface (no-slip) to the free-stream value $U$, because of viscosity. Outside it the flow is effectively inviscid.

Growth along a thin plate

 U ->   ------------------------------------------
        laminar         transition      turbulent
 ------------___________.........______________________
   delta ~ sqrt(x)                        delta ~ x^0.8
 ========================================= plate -->
 leading edge

Laminar flow (near the leading edge, $Re_x < 5\times10^5$)

• The layer starts with zero thickness at the leading edge and grows slowly as viscous effects spread outward.
• The fluid moves in smooth layers. Blasius's result: $\delta = \dfrac{5x}{\sqrt{Re_x}}$, so $\delta \propto \sqrt{x}$.
• The velocity profile is nearly parabolic, with a gentle gradient at the wall.

Turbulent flow (downstream, $Re_x > 5\times10^5$)

• After a transition region the flow becomes turbulent and eddies mix the fluid strongly.
• The layer is thicker and grows faster: $\delta = \dfrac{0.37x}{Re_x^{1/5}}$, $\delta \propto x^{0.8}$.
• The profile is fuller ($u/U = (y/\delta)^{1/7}$), the wall shear stress is larger, and a thin laminar sub-layer stays at the wall.

Here $Re_x = Ux/\nu$.

Two examples of use

1. Finding the skin-friction drag on ship hulls, aircraft wings and flat surfaces.
2. Designing streamlined shapes to delay separation and reduce drag, and wind tunnel sections using displacement thickness.