Boundary Layer Theory

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Development of Boundary Layer theory was an important event in the development of fluid mechanics. It was a significant step to numerous technological marvels such as passenger jets, fast submarine, etc.

For understanding the boundary layer theory we must first understand what boundary layer is.

Boundary Layer

Consider a section of length L of an infinite plate. A flowing stream with an initial maximum velocity U_eflows over the plate. Its flow velocity is zero at the point of contact with the plate/wall and varies to U_etill a certain distance δ from the plate. From 0 to L, the δ increases.

Beyond δ, the velocity of stream remains Ue. The locus of all these points (at a distance δ) Is called the boundary layer edge and the area below it is the Boundary Layer.

Strictly speaking, the value of δ is an arbitrary value because the friction force, depending on the molecular interaction between fluid and the solid body, decreases with the distance from the wall and becomes equal to zero at infinity.

In simple terms, the effects of friction on the stream due to the plate are not observed significantly beyond the boundary layer edge and the fluid flow is essentially free of viscosity.

Figure 1. Growth of a boundary layer on a flat plate.

It was L. Prandtl (1904) who proposed the fundamental concept of boundary layer which defines the boundary layer as a layer of fluid developing in flows with very high Reynold's number Re, that is with relatively low viscosity as compared with inertia forces. When bodies are exposed to a stream of air of high velocity or when bodies are very large compared to the air stream to which they are exposed. In this case, in a relatively thin boundary layer, friction shear stress (viscous shearing force): τ = η[∂u/∂y] (where η is the dynamic viscosity; u = u(y) – “profile” of the boundary layer longitudinal velocity component, Fig 1) may be very large; in particular, at the wall where u = 0 and τ_w = η[∂u/∂y]_w although the viscosity itself may be rather small.

The frictional forces are negligible and can be ignored outside the boundary layer(as compared with inertia forces), and on the basis of Prandtl’s concept, to consider two flow regions: the boundary layer where friction effects are large and the almost inviscid flow core. Making an assumption that boundary layer is very thin as compared to the length of the section i.e. δ << L, where L is the characteristic linear dimension of the section of body over which the flow occurs or the channel containing the flow, its thickness decreasing with growth of Re, fig.1), it is possible to estimate the one can estimate the order of magnitude of the boundary layer thickness from the following relationship:

Let’s understand this with an example. Consider an aeroplane which is flying at U_e= 400 km/hr, the boundary layer thickness at the wing trailing edge with one meter chord (profile length) is δ=0.015 m. According to experimentally established data, a laminar boundary layer is developed over the inlet section of the body. Gradually under some destabilizing factors, the boundary layer becomes unstable and transition of boundary layer to Turbulent Flow regime takes place. Experimental results have concluded the existence of a transition region between the turbulent and laminar regions. Sometimes, such as in case of high turbulence level of the external flow, the boundary layer becomes turbulent immediately downstream of the stagnation point of the flow. In case of some conditions, such as a severe pressure drop, an inverse phenomenon takes place in accelerating turbulent flows, namely flow relaminarization.

Fig. 3: Laminar and turbulent flow over a wing [2]

In spite of its relative thinness, the boundary layer is very significant for initiating processes of dynamic interaction between the floe and body. The boundary layer determines the aerodynamic drag and lift of the flying vehicle, or the energy loss for fluid flow in channels. In this case, a hydrodynamic boundary layer because there is also a thermal boundary layer which determines the thermodynamic interaction of heat transfer.

Computation of the boundary layer parameters is based on the solution of equations obtained from the Navier–Stokes equations for viscous fluid motion, which are first considerably simplified considering the thinness of the boundary layer. This can be considered to be the most significant role of the boundary layer theory.

Foundation of Boundary layer theory

The development of boundary layer theory was initiated by Ludwig Prandtl, presenting paper On the motion of a fluid with very small viscosity at the Third International Congress of Mathematics in August, 1904, at Heidelberg and published in the Proceedings of the Congress in the following year.

The equations of motion of a viscous fluid were established in the first half of the last century by Navier (1823), Poisson (1831), Saint-Venant (1843), and Stokes (1845), having attained the form that is now called the Navier-Stokes equations. Stokes used the equations to consider the small oscillations of a sphere in a viscous fluid by assuming that there is no slip, that is, no relative tangential velocity, at the surface of the sphere. The solutions obtained by Stokes, were confined to special cases, where it was possible to solve the Navier Stokes equations. To make the calculations easier, approximation to neglect viscosity was introduced but this lead to d'Alembert paradox, according to which a solid body of any shape placed in a uniform stream experiences no resistance.

Difficulties faced in the mathematical integration of equations of a viscous fluid made neglecting the nonlinear terms of the equation unavoidable. This approximation, justified only for slow motions, was made unavoidably also for faster motions, but with the optimistic hope that these solutions might give a better representation of the flow than those obtained by neglecting the viscosity. It was almost universally agreed that there is no slip at the solid wall in the case of slow motions. But in case of fast motion, there was division of views.

Pradtl’s paper

Pradtl recognized in his paper that the most important question concerning flow of fluid of small viscosity is the behaviour of the fluid at the wall of the solid boundary. He concluded that the the effects of viscosity are significant only within a thin transition layer, which is called the boundary layer. Beyond the boundary layer, the flow can be considered free of viscosity and is described by an irrotational motion to a high degree of accuracy. The small thickness of the boundary layer allows us to make certain approximations in the equations applicable for the region within the boundary layer, the variation of pressure normal to the wall is negligibly small, and the variation of velocity along the wall is much smaller than its variation normal to it. In case of two dimensional flow, since the effect moderate curvature of wall is insignificant due to it’s small value, x and y can be considered as the distances along and normal to the wall while u and v being the corresponding velocity components. the component of Navier Stokes equation is then simplified to

Where, t = time,

p = pressure,

ρ = density,

v = kinematical viscosity

Prandtl considered the solution of the equation for the simple case p = constant, that is, the case of a semi-infinite thin flat plate placed parallel to a stream of uniform velocity U, and obtained a rough estimate 1.1pvt/2[t/2U3/2 for the frictional resistance exerted on the two sides of unit width of a plate of length /. This was the first theoretical analysis of the frictional resistance, although the numerical coefficient 1.1 was later corrected by Blasius (1908) to 1.33.

Ludwig Pradtl at his water tunnel [3]

A remarkable consequence of the investigation from the standpoint of application was, according to Prandtl, that "in certain cases, the flow separates from the surface at a point entirely determined by external conditions. A fluid layer, which is set in rotation by the friction on the wall, is thus forced into the free fluid and, in accomplishing a complete transformation of the flow, plays the same role as the Helmholtz separation layers.

This was the early foundational work on Boundary layer Theory

Applications

Mathematically, application of the boundary - layer theory converts the character of governing Navier-Stroke equations from elliptic to parabolic
This allows the marching in flow direction, as the solution at any location is independent of the conditions farther downstream.
The parameters that describe the flow within the boundary layer are important in many areas such as:

Wing stall
The skin friction drag on an object
The heat transfer that occurs in high speed flight.

Limitations of Boundary Layer theory

The concept of boundary layer is important for understanding fluid flows with higher values of Reynold’s number (Re>>1). But in the use of boundary layer as a quantitative asymptotic theory, we face severe limitations in the concepts of turbulence and boundary layer separation.

We can apply boundary layer theory when value of Reynold’s no. approaches infinity. But in this limit, boundary layer flows are turbulent. Turbulent flows have a complex and random fine structure and their length scales are dependent on Re. When Reynold’s no. is of the order 10⁶, the boundary layer becomes turbulent. The phenomenon where the boundary layer exits the boundary and enters the interior of the fluid is called boundary layer separation. In such cases, a strong coupling between the outer flow and the boundary layer comes into picture which is accompanied by turbulence. A systematic mathematical theory for turbulent flows is still undeveloped.

References:

1. https://www.thermopedia.com/content/595/

2. Itiro Tani, History of boundary layer theory, National Aerospace Laboratory, 1880 Jindaiji, Chofu, Tokyo, Japan

3. https://www.grc.nasa.gov/www/k-12/airplane/boundlay.html

4. https://nptel.ac.in/content/storage2/courses/112104118/lecture-28/28-4_boundary_layer_condn.htm

5. https://www.math.ucdavis.edu/~hunter/m204/boundary.pdf

Image sources:

1. https://www.sciencephoto.com/media/353466/view/typhoon-fighter-plane-aerodynamics

2. http://www.pilotfriend.com/training/flight_training/aero/boundary.htm

3. https://www.researchgate.net/profile/Michael-Eckert-5/publication/51918331/figure/fig1/AS:655137915826178@1533208459251/Ludwig-Prandtl-at-his-water-tunnel-in-the-mid-to-late-1930s-Reproduction-from-the.png