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The Flow of Viscous Fluids. Flow in the Vicinity of a Wall: Boundary Layers and Films

1.1. Introduction

An important class of problems concerns the interaction between a viscous flow and a boundary: an open solid boundary (plate) or a closed one (tube), a fluid at rest, another flow or another fluid.

Given their common characteristics, these flows can be grouped in the category of boundary layers:

• - Outer boundary layers: interaction between a flow of indefinite extension and a flat, curved, cylindrical or other shaped wall
• - Inner boundary layers: interaction between a flow of finite extension and a partially or fully closed wall: pipe flow, open channel flow
• - "Mixing" boundary layers: jets, etc.

These flows can be laminar or turbulent. In this respect, viscosity plays a key role.

1.2. Characteristics and classification of boundary layers

In essence, boundary layers are "thin" flows. Therefore, in any boundary layer, a distinction between "lateral" and "longitudinal" scales is established. In particular, the gradients of parameters (velocity, temperature, concentration) are much stronger along the lateral dimensions than along the longitudinal ones.

In the case of outer boundary layers, a two-step calculation method is applicable, which has already been mentioned in Chapter 3 of [LED 17]:

1. a) When a solid body is placed in a uniform flow, the so-called "potential" flow is calculated (see the kinematics review) based on a perfect inviscid fluid theory.
2. b) Viscosity in the "thin" boundary layer is then taken into account.

The "inner" boundary layers are present in areas where the regime sets in. Therefore, the scales to be compared are the radius of a pipe relative to the length L of this inflow area.

The jet boundary layers relate the lateral dimension of a jet, expressed by a lateral profile of velocity, and the longitudinal dimension of the jet.

The analytical approaches in this manual are focused on the outer or inner boundary layer. The numerical approach is better suited for jet problems.

A prototype of outer boundary layers consists of the flow developed by a flat plate in the uniform flow of velocity Ue [or locally uniform flow if Ue = Ue(x) has an axial variation]. In this case, there is a non-disturbed flow (or "potential" flow; this problem has been mentioned in Chapter 3 of [LED 17]) and a border layer flow, where the flow connects to the wall area. This connection area with thickness d is "thin" compared with the longitudinal dimension L of the body being considered.

There are longitudinal scales (along Ox in Cartesian coordinates), which are always larger than the lateral scales (along Oy in Cartesian coordinates).

Or expressed in orders of magnitude:

There are two consequences for the orders of magnitude of the terms in the equations of fluid mechanics:

The boundary layer is "thin"; therefore, the flow is "nearly parallel to the flow". In Cartesian coordinates and for a plane flow, a first relation can be established between the longitudinal component u and lateral component v of velocity: u <<v.

NOTE.- As will be seen, to ensure continuity, this cannot be rigorously valid. This will particularly be the case when employing integral methods. It will be used in certain cases as boundary "property" to obtain approximate results: see, for example, Stokes' first problem presented further on.

There is a similar relation, in terms of order of magnitude, between longitudinal and lateral derivatives of the same function f:

A simplification of the equations of fluid mechanics is thus possible, though to a limited extent.

Indeed, in orders of magnitude:

These three terms have the same order of magnitude.

For a plane incompressible steady flow above a flat plate, we have:

It is worth noting that not all the terms containing the derivative with respect to x disappear, as these equations contain products of small terms and large terms .

A physicist's approach to these order of magnitude-related notions is recommended. In the outer boundary layers, when the distance from the wall increases, velocity reaches quite rapidly a value close to that of the potential flow, Ue. Rigorously speaking, it is an asymptotic variation, meaning that u tends to Ue, when x tends to infinity. In wind tunnel, there would be infinities of several centimeters!

The thickness of the boundary layer, although denoted by d, is not infinitely small. In wind tunnel applications, for a plate with length of several tens of centimeters, d is also measured in centimeter. Given a ship with length L = 250m, d is measured in meter. In a geographical area, the atmospheric boundary layer can measure 1-2 km!

Moreover, an infinitely small d compared to L often comes down to ratios of the order of 10-2.

1.2.1. Boundary layers - various approaches

For laminar flows, the boundary layer equations can be solved. Several examples, such as the Blasius theory and the Stokes theory, will be given below.

There are various approaches to turbulent flows: semi-empirical theories, the mixing length theory, the Boussinesq viscosity models and numerical approaches (Reynolds decomposition, k - e methods). References to the very rich literature on this subject will be provided.

Reasonable results can also be achieved with "lower costs", by using the integral methods. Although quite old, this approach is still worth being known. It was our intention to initiate the interested reader into these techniques, and a specific paragraph is dedicated to this purpose.

1.3. The outer boundary layers: an analytical approach

1.3.1. The laminar boundary layer developed by a flat plate in a uniform flow

Although laminar flows may be perceived as rather academic from a practical point of view, the reasoning developed in regards to this model is highly inciting and formative for the thinking process of fluid mechanics scientists.

Let us consider a fluid with constant physical properties (density µ and kinematic viscosity ?). The potential flow is then obviously a uniform flow of velocity Ue. We shall consider the case of a plane steady flow. An orthonormal frame of reference is attached to the system. The origin is on the leading edge of the plate; x and y axes are parallel and perpendicular to the plate, respectively.

The velocity vector verifies the following equations:

Continuity equation:

Impulse equation:

The term containing calls for two remarks:

The flow is not rigorously parallel in the boundary layer. In effect, v(x, y) is smaller than u(x, y), but not rigorously null. This makes physical sense: the plate "slows down" the fluid, which is "chased away from the plate" and into the free flow. Nevertheless, it is close enough to a parallel flow to allow us to consider that perpendicularly to these flow lines, and therefore to the plate, the lateral gradient of is null.

A "full" d (full derivative) with respect to x is justified here. Similarly, the value of pG(x) will be to the same as the one observed for the same coordinate x in the free flow.

From a physical perspective, Ue can vary if a wind tunnel of variable cross-section is considered, for example. In the potential flow, where the fluid is deemed perfect, Bernoulli's theorem is verified. Therefore:

which offers the possibility to replace with

When the potential flow is rigorously uniform, this term is null, a case which will be considered in what follows.

The Blasius problem

Blasius solved the problem of the laminar boundary layer of a fluid with constant physical properties over a flat plate of indefinite width placed in the path of wind of a uniform flow.

The system of equations that relate the two unknowns u(x, y) and u(x, y) becomes

The boundary conditions being written

It is worth noting that the problem states an asymptotic variation of velocity toward the outer flow "to infinity". In fact, rapid convergence will be noted. A boundary layer thickness is defined by the point in which u(x,d) = 0,99Ue. This entails d = d(x). We shall verify in the following...