# Fluid dynamic boundary layer

The fluid dynamic boundary layer , often also simply called “ boundary layer ”, describes the area in a flowing fluid on a wall in which the viscosity of the fluid exerts an influence on the velocity profile perpendicular to the wall.

With laminar flow and a sufficiently large Reynolds number , the viscosity of the fluid can be neglected in the majority of a flow field. However, the influence of the viscosity in the boundary layer is by no means negligible. The thickness of this boundary layer is very small when the flow is present, but the shear stress resistance of the body in flow forms in it , which together with the pressure resistance makes up the entire flow resistance of a body. Boundary layer detachments , after which the main flow no longer follows the wall, often have undesirable, sometimes dramatic effects. Knowledge of the behavior of the fluid dynamic boundary layers is therefore important for construction in aircraft construction ( wings ), shipbuilding (flow around the ship's hull, rudder and propeller blades), automobile construction ( c w value ), wind power stations and turbine construction (turbine blades).

## Boundary layer theory

Boundary layer theory is a field of fluid mechanics that deals with fluid movement near walls with very little friction. The boundary layer theory was introduced by Ludwig Prandtl in 1904 during a lecture at the Heidelberg Mathematicians Congress. Prandtl divided the flow around a body into two areas:

1. an external flow in which the viscous friction losses can be neglected, and
2. a thin layer ("boundary layer") near the body, in which the viscous terms from the Navier-Stokes equations (momentum equations) are taken into account.

The conditions in the boundary layer make it possible to considerably simplify the Navier-Stokes equations , which describe the flow of air and water realistically, and the resulting so-called boundary layer equations can even be solved analytically . As a result of the simplifications, the calculation effort for bodies in a flow (e.g. aircraft, cars or ships) is considerably reduced.

## Boundary layer for small Reynolds numbers

For sufficiently small Reynolds numbers, the fluid dynamic boundary layer is laminar and the main flow is in the same direction. The special conditions in the boundary layer allow the pressure gradient perpendicular to the wall to be neglected: the pressure is approximately constant over the thickness of the boundary layer and is impressed by the main flow. Furthermore, the change in speed in the direction parallel to the wall compared to that perpendicular to the wall can be disregarded. Application of these assumptions in the Navier-Stokes equations leads to the boundary layer equations mentioned above .

### Boundary layer thickness

Formation of a laminar boundary layer under the blue line on a flat surface (lower horizontal line). Re x = v 0 x / ν is here for each x smaller than Re crit = 5 · 10 5 .

A flat plate overflowing in the direction is considered, as in the picture, where the direction is perpendicular to the plate. The fluid elements adhere directly to the body ( adhesion condition :) and within the boundary layer their speed adapts to the speed of the main flow. Since the speed can theoretically never reach the ambient speed of the flow due to viscous friction alone, the thickness of this "speed boundary layer " is defined as reaching 99% of the ambient speed : ${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle v_ {x} (y = 0) = 0}$${\ displaystyle v_ {0}}$${\ displaystyle \ delta}$${\ displaystyle v_ {0}}$

${\ displaystyle \ delta \ colon \ quad v_ {x} (y = \ delta) = 0 {,} 99 \ cdot v_ {0}}$

In the case of a flat panel as in the picture, the boundary layer thickness increases with the root of the distance to the front edge of the panel:

${\ displaystyle \ delta \ simeq {\ sqrt {\ frac {\ nu \ cdot x} {v_ {0}}}} \ ,.}$

This is the kinematic viscosity (dimension L 2 T −1 , unit / s ) of the fluid. ${\ displaystyle \ nu}$

Other measures of the thickness of the boundary layer are

1. the displacement thickness δ * or δ 1 , which indicates how far the main flow is pushed away from the body wall by the boundary layer, and
2. the momentum loss thickness θ or δ 2 , which represents the reduction in momentum current in the boundary layer due to friction .

### Speed ​​profile

From the wall to the end of the boundary layer ( ) the velocity profile corresponds approximately to a quadratic function : ${\ displaystyle 0 \ leq y \ leq \ delta}$

${\ displaystyle v_ {x} (y) \ approx v_ {0} \ cdot \ left [1- \ left (1 - {\ frac {y} {\ delta}} \ right) ^ {2} \ right]}$

In the direction of flow, the thickness of the fluid dynamic boundary layer increases (blue curve in the figure). In pipes or channels, the boundary layers can grow together from the edges, so that the laminar flow is fully developed and the speed is a parabolic function of the distance from the wall.

### Reynolds numbers

The Reynolds number , which is important in the boundary layer flow, increases with the boundary layer thickness

${\ displaystyle Re _ {\ delta}: = {\ frac {v_ {0} \ delta} {\ nu}}}$

educated. The Reynolds number

${\ displaystyle Re_ {l}: = {\ frac {v_ {0} l} {\ nu}}}$

increases with the length of the flow along the wall and it results: ${\ displaystyle l}$

${\ displaystyle {\ frac {\ delta} {l}} \ simeq {\ frac {1} {l}} {\ sqrt {\ frac {\ nu l} {v_ {0}}}} = {\ frac { 1} {\ sqrt {Re_ {l}}}} \ quad {\ text {and}} \ quad Re _ {\ delta} \ simeq {\ frac {v_ {0}} {\ nu}} {\ sqrt {\ frac {\ nu l} {v_ {0}}}} = {\ sqrt {Re_ {l}}} \ ,.}$

After a certain length of travel along a plate, the boundary layer becomes unstable and turns into a turbulent state, see below. This happens with the critical Reynolds number

${\ displaystyle Re _ {\ text {krit}} = \ left. {\ frac {v_ {0} l} {\ nu}} \ right | _ {\ text {krit}} = 5 \ cdot 10 ^ {5} \ ,.}$

The Reynolds number can be higher with a low-disturbance flow.

### Wall shear stress

Due to the viscosity, the fluid transfers momentum to the wall, which is noticeable through the wall shear stress τ w . In Newtonian fluids this is of the order of magnitude

${\ displaystyle \ tau _ {w} = \ eta \ left. {\ frac {\ partial v_ {x}} {\ partial y}} \ right | _ {y = 0} \ simeq \ eta {\ frac {v_ {0}} {\ delta}} \ simeq \ eta {\ frac {v_ {0}} {\ sqrt {\ frac {\ nu l} {v_ {0}}}}}} = {\ sqrt {\ frac { \ rho \ eta v_ {0} ^ {3}} {l}}}}$

The parameter η = ρ · ν is the dynamic viscosity (dimension ML –1 T −1 , unit Pa s ). If b is the width and l is the length of a plate with a flow around both sides, then the following applies to its shear stress resistance

${\ displaystyle W = 2bl \ tau _ {w} \ simeq b {\ sqrt {\ rho \ eta v_ {0} ^ {3} l}} \ ,.}$

### Measure of time

Fluid elements that do not flow too close to the wall remain near the wall for a time proportional to . The following applies: ${\ displaystyle l / v_ {0}}$

${\ displaystyle \ delta \ simeq {\ sqrt {\ frac {\ nu l} {v_ {0}}}} \ simeq {\ sqrt {\ nu t}} \ ,.}$

This equation shows that the boundary layer thickness increases proportionally to the square root of the time at the beginning of the movement . ${\ displaystyle t}$

## Boundary layer for large Reynolds numbers

Transition from laminar to turbulent boundary layer with a plate flow

At high Reynolds numbers, the boundary layer flow is turbulent ; H. within the boundary layer, the fluid elements can move in any direction. The thickness of the boundary layer is thicker than that of a laminar boundary layer, but remains very limited, see picture.

The main flow is said to be laminar. Then after a certain length of the flow along the wall, the flow becomes unstable. The Tollmien-Schlichting waves , named after their discoverers, are formed , whose wave front initially runs perpendicular to the direction of flow but parallel to the wall. Downstream these waves are superimposed with three-dimensional waves (indicated by helical lines), whereby the wave front adopts a sawtooth shape and characteristic Λ-vortices arise. These Λ-eddies disintegrate into turbulence spots (small helical lines), which eventually grow together to form a turbulent boundary layer. The turbulence improves the exchange of momentum between the fluid elements, which is why they have a higher mean wall-parallel velocity near the wall than in the laminar boundary layer. ${\ displaystyle {\ bar {v}} _ {x}}$

## Viscose underlayer

Each wall has - even under a turbulent boundary layer - a viscous sublayer ( English viscous sublayer ), formerly laminar sublayer mentioned. Only when the roughness of the wall penetrates this sub-layer does it have an impact on the boundary layer, the frictional resistance and the flow. If, on the other hand, the sub-layer completely covers the roughness, the wall is hydraulically smooth .

In the viscous lower layer, the dimensionless coordinate formed with the wall shear stress velocity is smaller than one ${\ displaystyle u _ {\ tau}}$

${\ displaystyle y ^ {+}: = {\ frac {u _ {\ tau}} {\ nu}} \ cdot y <1}$

and the time-averaged speed in the thin sublayer increases approximately linearly with the distance to the wall:

${\ displaystyle {\ bar {v}} _ {x} = u _ {\ tau} \ cdot y ^ {+} = {\ frac {\ tau _ {w}} {\ eta}} \ cdot y \ ,. }$

## Construction engineering

The boundary layer is important for the thermal insulation of buildings. Stone and glass windows have a high thermal conductivity compared to air. In fact, there is a boundary layer of air on the walls, which with λ = 5.8 * 10 −5 insulates better than the masonry. The boundary layer can become thinner during a storm.

## literature

• H. Oertel (ed.): Prandtl guide through fluid mechanics . Fundamentals and phenomena. 13th edition. Springer Vieweg, 2012, ISBN 978-3-8348-1918-5 .
• H. Schlichting, K. Gersten: boundary layer theory . 9th edition. Springer Verlag, Berlin 1997, ISBN 3-540-55744-X .
• Ernst Götsch: Aircraft technology . Motorbuchverlag, Stuttgart 2003, ISBN 3-613-02006-8
• OA Olejnik, VN Samokhin: Mathematical models in boundary layer theory. Chapman & Hall / CRC, Boca Raton 1999, ISBN 1-58488-015-5
• Joseph A. Schetz: Boundary layer analysis . Prentice-Hall, Englewood Cliffs 1993, ISBN 0-13-086885-X