# boundary-layer flow

## Boundary-layer flow

That portion of a fluid flow, near a solid surface, where shear stresses are significant and the inviscid-flow assumption may not be used. All solid surfaces interact with a viscous fluid flow because of the no-slip condition, a physical requirement that the fluid and solid have equal velocities at their interface. Thus a fluid flow is retarded by a fixed solid surface, and a finite, slow-moving boundary layer is formed. A requirement for the boundary layer to be thin is that the Reynolds number of the body be large, 10^{3} or more. Under these conditions the flow outside the boundary layer is essentially inviscid and plays the role of a driving mechanism for the layer. *See* Reynolds number

A typical low-speed or laminar boundary layer is shown in the illustration. Such a display of the streamwise flow vector variation near the wall is called a velocity profile. The no-slip condition requires that *u*(*x*, 0) = 0, as shown, where *u* is the velocity of flow in the boundary layer. The velocity rises monotonically with distance *y* from the wall, finally merging smoothly with the outer (inviscid) stream velocity *U*(*x*). At any point in the boundary layer, the fluid shear stress &tgr; is proportional to the local velocity gradient, assuming a newtonian fluid. The value of the shear stress at the wall is most important, since it relates not only to the drag of the body but often also to its heat transfer. At the edge of the boundary layer, &tgr; approaches zero asymptotically. There is no exact spot where &tgr; = 0; therefore the thickness δ of a boundary layer is usually defined arbitrarily as the point where *u* = 0.99*U*. *See* Laminar flow

When a flow enters a duct or confined region, boundary layers immediately begin to grow on the duct walls. An inviscid core accelerates down the duct center, but soon vanishes as the boundary layers meet and fill the duct with viscous flow. Constrained by the duct walls into a no-growth condition, the velocity profile settles into a fully developed shape which is independent of the streamwise coordinate. The pressure drops linearly downstream, balanced by the mean wall-shear stress. This is a classic and simple case of boundary-layer flow which is well documented by both theory and experiment.

A classic incompressible boundary-layer flow is a uniform stream at velocity *U*, moving past a sharp flat plate parallel to the stream. In the Reynolds number range 1 × 10^{3} to 5 × 10^{5}, the flow is laminar and orderly, with no superimposed fluctuations. The boundary-layer thickness δ grows monotonically with *x*, and the shape of the velocity profile is independent of *x* when normalized. The profiles are said to be similar, and they are called Blasius profiles.

The Blasius flat-plate flow results in closed-form algebraic formulas for such parameters as wall-shear stress and boundary-layer thickness as well as for temperature and heat-transfer parameters. These results are useful in estimating viscous effects in flow past thin bodies such as airfoils, turbine blades, and heat-exchanger plates.

The flat plate is very distinctive in that it causes no change in outer-stream velocity *U*. Most body shapes immersed in a stream flow, such as cylinders, airfoils, or ships, induce a variable outer stream *U*(*x*) near the surface. If *U* increases with *x*, which means that pressure decreases with *x*, the boundary layer is said to be in a favorable gradient and remains thin and attached to the surface. If, however, velocity falls and pressure rises with *x*, the pressure gradient is unfavorable or adverse. The low-velocity fluid near the wall is strongly decelerated by the rising pressure, and the wall-shear stress drops off to zero. Downstream of this zero-shear or separation point, there is backflow and the wall shear is upstream. The boundary layer thickens markedly to conserve mass, and the outer stream separates from the body, leaving a broad, low-pressure wake downstream. Flow separation may be predicted by boundary-layer theory, but the theory is not able to estimate the wake properties accurately.

In most immersed-body flows, the separation and wake occur on the rear or lee side of the body, with higher pressure and no separation on the front. The body thus experiences a large downstream pressure force called pressure drag. This happens to all blunt bodies such as spheres and cylinders and also to airfoils and turbomachinery blades if their angle of attack with respect to the oncoming stream is too large. The airfoil or blade is said to be stalled, and its performance suffers.

All laminar boundary layers, if they grow thick enough and have sufficient velocity, become unstable. Slight disturbances, whether naturally occurring or imposed artificially, tend to grow in amplitude, at least in a certain frequency and wavelength range. The growth begins as a selective group of two-dimensional periodic disturbances, called Tollmien-Schlichting waves, which become three-dimensional and nonlinear downstream and eventually burst into the strong random fluctuations called turbulence. The critical parameter is the Reynolds number. The process of change from laminar to turbulent flow is called transition.

The turbulent flow regime is characterized by random, three-dimensional fluctuations superimposed upon time-mean fluid properties, including velocity, pressure, and temperature. The fluctuations are typically 3–6% of the mean values and range in size over three orders of magnitude, from microscale movements to large eddies of size comparable to the boundary-layer thickness. They are readily measured by modern instruments such as hot wires and laser-Doppler velocimeters. *See* Anemometer

The effect of superimposing a wide spectrum of eddies on a viscous flow is to greatly increase mixing and transport of mass, momentum, and heat across the flow. Turbulent boundary layers are thicker than laminar layers and have higher heat transfer and friction. The turbulent mean-velocity profile is rather flat, with a steep gradient at the wall. The edge of the boundary layer is a ragged, fluctuating interface which separates the nonturbulent outer flow from large turbulent eddies in the layer. The thickness of such a layer is defined only in the time mean, and a probe placed in the outer half of the layer would show intermittently turbulent and nonturbulent flow.

As the stream velocity *U* becomes larger, its kinetic energy, *U*^{2}/2, becomes comparable to stream enthalpy, *c _{p}T*, where

*c*is the specific heat at constant pressure and

_{p}*T*is the absolute temperature. Changes in temperature and density begin to be important, and the flow can no longer be considered incompressible. Liquids flow at very small Mach numbers, and compressible flows are primarily gas flows.

*See*Mach number

In a flow with supersonic stream velocity, the no-slip condition is still valid, and much of the boundary-layer flow near the wall is at low speed or subsonic. The fluid enters the boundary layer and loses much of its kinetic energy, of which a small part is conducted away although most is converted into thermal energy. Thus the near-wall region of a highly compressible boundary layer is very hot, even if the wall is cold and is drawing heat away. The basic difference between low and high speed is the conversion of kinetic energy into higher temperatures across the entire boundary layer.

In a low-speed (incompressible) boundary layer, a cold wall simply means that the wall temperature is less than the free-stream temperature. The heat flow is from high toward lower temperature, that is, into the wall. For a low-speed insulated wall, the boundary-layer temperature is uniform. For a high-speed flow, however, an insulated wall has a high surface temperature because of the viscous dissipation energy exchange in the layer.

Except for the added complexity of having to consider fluid pressure, temperature, and density as coupled variables, compressible boundary layers have similar characteristics to their low-speed counterparts. They undergo transition from laminar to turbulent flow but typically at somewhat higher Reynolds numbers. Compressible layers tend to be somewhat thicker than incompressible boundary layers, with proportionally smaller wall-shear stresses. They tend to resist flow separation slightly better than incompressible flows.

In a supersonic outer stream, shock waves can always occur. Shocks may form in the boundary layer because of obstacles in the layer or downstream, or they may be formed elsewhere and impinge upon a boundary. In either case, the pressure rises sharply behind the shock, an adverse gradient, and this tends to cause early transition to turbulence and early flow separation. Special care must be taken to design aerodynamic surfaces to accommodate or avoid shockwave formation in transonic and supersonic flows. *See* Compressible flow

As boundary layers move downstream, they tend to grow naturally and undergo transition to turbulence. Boundary layers encountering rising pressure undergo flow separation. Both phenomena can be controlled at least partially. Airfoils and hydrofoils can be shaped to delay adverse pressure gradients and thus move separation downstream. Proper shaping can also delay transition. Wall suction removes the low-momentum fluid and delays both transition and separation. Wall blowing into the boundary layer, from downward-facing slots, delays separation but not transition. Changing the wall temperature to hotter for liquids and colder for gases delays transition. Practical systems have been designed for boundary-layer control, but they are often expensive and mechanically complex. *See* Airfoil, Streamlining, Viscosity