Wall Shear Stress - 1 | Fluid Mechanics for Mechanical Engineering PDF Download

• For engineering applications we often prefer approximate integral methods which give sufficiently accurate answers without solving Prandtl's boundary-layer differential equations exactly.
• Kármán and Pohlhausen developed a convenient method that satisfies the boundary conditions and integral conservation laws rather than the full differential equations. The method leads to the momentum integral equation.
• Consider steady, two-dimensional, incompressible flow and integrate the dimensional momentum equation in the streamwise direction from the wall (y = 0) to y = δ (the free-stream interface of the boundary layer). The integrated form is:

(29.10)

• The second term on the left of (29.10) may be expanded as follows.

(29.11)

• Substituting Eq. (29.11) into Eq. (29.10) gives the next integral relation.

(29.12)

• Substituting the relation between the local outer (inviscid) velocity and the free stream velocity U_∞ in Eq. (29.12) yields

which can be reduced to the standard momentum integral form shown next.

• Since the integrands vanish outside the boundary layer, the upper integration limit may be extended to infinity (that is, δ → ∞) when using similarity forms.

(29.13)

• The term representing dU_∞/dx (space-wise acceleration of the free stream) signifies the presence of a pressure gradient in the flow direction. A non-zero value indicates that the outer inviscid flow is accelerating or decelerating along x.
• For external flows the sign and magnitude of dU_∞/dx depend on the body shape. In entrance regions of internal flows (pipes, channels) a finite dU_∞/dx can exist.
• For the classical flat plate in a uniform stream U_∞ is constant so dU_∞/dx = 0 and the momentum integral equation reduces to the simpler form:

(29.15)

Separation of the Boundary Layer

• Under certain conditions the flow adjacent to the wall reverses direction; this phenomenon is called boundary-layer separation.
• Separation occurs because the boundary layer loses too much momentum near the wall while attempting to move downstream against an increasing pressure (that is, when the pressure gradient in the flow direction is adverse, ∂P/∂x > 0).
• Figure 29.2 illustrates flow past a circular cylinder in an unbounded medium and the development of the wake behind the cylinder.

Consider flow around a cylinder: up to θ = 90° the streamlines behave like those in a nozzle (favourable pressure gradient); beyond θ = 90° the flow area diverges and the outer flow experiences an adverse pressure gradient.

1. Up to θ = 90°, the pressure gradient is negative (favourable) and the pressure force and the inertial (streamwise acceleration) force act in the same direction.
2. Beyond θ = 90°, the pressure gradient becomes positive (adverse) and the pressure force opposes the inertial force in the inviscid zone.

• In the viscous region close to the wall, viscous forces oppose the combined pressure and inertial forces. For θ ≤ 90° the viscous forces are overcome by conversion of pressure energy into kinetic energy and the boundary layer remains attached.
• For θ > 90° the adverse pressure increases; viscous and pressure forces together may overcome the forward inertial force and the near-wall fluid particles may decelerate, stop and eventually reverse direction. Where reversal first appears is the point of separation and a broad, often unsteady wake develops downstream.

Fig. 29.2 Flow separation and formation of wake behind a circular cylinder

1. Until θ = 90° the pressure force and the streamwise acceleration act together (pressure gradient favourable).
2. Beyond θ = 90° the pressure gradient is adverse; pressure and inertial forces oppose each other in the inviscid region.

When viscous effects are included near the wall:

1. Up to θ = 90° the viscous force opposes the combined pressure and inertial forces, but the fluid overcomes viscous resistance due to the continuous conversion of pressure head into kinetic energy.
2. Beyond θ = 90° the viscous zone experiences both adverse pressure and viscous opposition, which may lead to deceleration and eventual flow reversal near the wall.

• Depending on the magnitude of the adverse pressure gradient, separation typically occurs around θ ≈ 90° (in experiments for a cylinder it is observed near θ ≈ 81°). The separated near-wall layers are carried into the wake where vortices and eddies form.
• The point of separation can be identified mathematically as the streamwise location where the wall shear stress becomes zero, i.e. t_w = 0, and beyond which reverse flow occurs.

(29.16)

At the separation point the shear stress at the wall t_w = 0. Downstream of this point the adverse pressure causes backflow in the near-wall region.

• Separation may also be understood by examining the second derivative of the streamwise velocity u with respect to y at the wall. From the dimensional momentum equation at the wall (where u = v = 0) one obtains the expression below.

(29.17)

• Under a favourable pressure gradient (dP/dx < 0)="" we="" have="" the="" following="" consequences="" for="" the="" curvature="" of="" the="" velocity="">

1. 
   (From Eq. (29.17)).
2. As y increases toward the free stream the velocity u approaches U_∞ asymptotically, so ∂u/∂y decreases with y.
3. Thus ∂²u/∂y² remains negative near the edge of the boundary layer and the curvature of the velocity profile is negative (concave downward) as shown in Fig. 29.3a.

• Under an adverse pressure gradient the curvature near the wall must become positive (since ∂P/∂x > 0 implies positive ∂²u/∂y² at the wall), while near the edge of the boundary layer the curvature is still negative. Therefore a point of inflection exists somewhere within the boundary layer where ∂²u/∂y² = 0. This point of inflection is a precursor to separation, but separation itself occurs when the shear at the wall vanishes, i.e. ∂u/∂y|_wall = 0.

1. At the wall, the curvature ∂²u/∂y² must be positive under adverse pressure gradient (see
   
   ).
2. Near the boundary layer edge the curvature returns to negative values (see
   
   and
   
   ), so there must be a point with ∂²u/∂y² = 0 - the point of inflection (see
   
   = 0).
3. Separation requires ∂u/∂y|_wall = 0 (see
   
   = 0 at the wall).
4. The wall curvature sign
   
   is consistent with separation occurring only under adverse pressure gradient; at the edge of the boundary layer we have
   
   = 0. It therefore follows that if separation occurs there must exist an inflection point in the velocity profile.

(a) Favourable pressure gradient,

(b) Adverse pressure gradient,

1. Returning to the cylinder wake: the inviscid potential flow pressure distribution (shown by the solid line in earlier figures) predicts symmetric pressures. However, experimentally the boundary layer detaches near θ ≈ 81° (rather than 90°), and the separated wake maintains a pressure close to the separation-point pressure.
2. In the wake the eddies and rotational flow prevent efficient conversion of rotational kinetic energy into pressure head. The actual pressure distribution thus departs from the inviscid prediction (indicated by a dotted line in figure). The wake pressure is lower than the forward stagnation pressure and this pressure difference produces a net drag on the body.
3. The drag arising from pressure difference between forward and wake regions is called form drag. The drag due to shear stress acting on the body surface is called skin friction drag. The total aerodynamic or hydrodynamic drag is generally the sum of form drag and skin friction drag.