1.1.1 Navier-Stokes equation, Hagen-Poiseuille
The first person asking these basic questions was Osborne Reynolds in the 19th century.
What was known at that time is the Navier-Stokes equation, which correctly describes the motion of fluids. Prior to it, Euler (1752) first applied Newton’s second law to fluids and derived an equation describing the motion of frictionless (inviscid) fluids. These equations are still widely used in aerodynamics and astrophysics. However, to describe flows close to walls and the onset of turbulence Euler’s equations are not satisfactory. In such cases the internal friction of the fluid, characterized by its viscosity is of tremendous importance. Navier (1823) and Stokes (1845) combined in their equation for the first time the viscous force of the internal motion of the fluid with the Euler equation. As a boundary condition they assumed that the fluid has zero velocity where it is in contact with a stationary wall (‘no-slip boundary condition’). Although the final form of the equation and boundary conditions were correctly derived by Stokes in 1845, it took much longer to verify its validity experimentally (see Section 1.2.2).
Figure 1.1: Hagen (1839) and Poiseuille (1840) analytically obtained the velocity profile of laminar pipe flow, which is parabolic.
The Navier-Stokes equation reads ρ ∂~u
∂t +(~u· ∇)~u
!
=−∇p+µ∇2~u+F,~ ∇ ·~u=0,
where the first equation expresses the balance of linear momentum and the second con-servation of mass. Here~u is the velocity field, p the pressure,ρ and µthe fluid density and dynamic viscosity, andF~an external force.
The Navier-Stokes equation is a nonlinear partial differential equation, which makes it very difficult to solve it. Mathematically it is not even clear, if a three dimensional solution always exists and if so, if it contains no singularities. A price of US$ 1.000.000 is set by the Clay Mathematics Institute to solve this problem, which is one out of seven millennium prize problems (Carlsonet al.2006).
Only very few exact solutions of the Navier-Stokes equation are known. The perhaps most relevant one describes the parabolic velocity profile of a flow through a straight, circular pipe, as shown in Fig. 1.1. The flow assumes such a profile only after a certain distance from the pipe entrance, which is inversely proportional to the viscosity (Tietjens 1970). When the fluid enters the pipe from e.g. a large container, the streamwise velocity is nearly uniform across the bulk of the pipe. However, the no-slip boundary condition enforces zero-velocity at the pipe wall, which leads to a spatially developing boundary layer. While the fluid is flowing downstream, the boundary layer thickness increases due to viscosity. Once the effect of the viscosity has reached the centerline of the pipe, the velocity profile becomes parabolic. The corresponding volume fluxQis proportional to the applied pressure difference and was obtained by Hagen (1839) and Poiseuille (1840):
Volume fluxQ= (p0− p1)R4
8µl ,
with (p0−p1) being the pressure drop between two positions at distancelalong the pipe,R the radius of the pipe andµthe dynamic viscosity of the fluid. However, this linear relation is only valid for laminar flows as in the capillary pipes of Poiseuille. In larger pipes Hagen (1854)1and also Darcy (1857)2observed the onset of disordered motion (e.g. turbulence)
1mentioned inTietjens (1970)
2mentioned in Mullin (2011)
when the flow velocity was increased. In this case, the Hagen-Poiseuille law was not valid any more. This is to my knowledge the first time that the dynamics of flows was divided in two different classes by flow visualization. The first one is the laminar flow, here the fluid is moving in nearby layers, no mixing between these layers is observed and the flow velocity is often constant with time. The other type is turbulent flow, here many vortices appear with different sizes, interacting in an unpredictable way, thereby mixing the fluid.
In turbulent flows the friction of the flow is substantially increased, which is exactly what Hagen and Darcy observed in their pipes.
1.1.2 Reynolds pioneering experiments
But why the linear dependence of the friction with the fluid velocity is not valid any more when turbulence sets in was not clear at all.
‘This accidental fitness of the theory to explain certain phenomena while en-tirely failing to explain others, affords strong presumption that there are some fundamental principles of fluid motion of which due account has not been taken in the theory’ (Reynolds 1883).
Reynolds revealed this‘fundamental principles’with his pioneering, extremely care-fully conducted pipe flow experiments. He used straight, smooth pipes made of glass with different diameters. The working fluid was water and the flow rate and temperature were accurately controlled. In addition he was able to visualize the flow by injecting ink at the entrance of the pipe center. In this experiment he could confirm the onset of turbulence, when the fluid velocity was increased above a critical value (see Fig. 1.3). A systematic continuation of this experiment for different pipe diameters and temperatures lead him to the conclusion, that
‘the general character of the motion of fluids [. . . ] depends on the relation between a physical constant of the fluid [the viscosity] and the product of the linear dimensions of the space occupied by the fluid and the velocity’
(Reynolds 1883).
This non-dimensional parameter became much later known as the Reynolds number Re(Sommerfeld 1908, Blasius 1911, von Kármán 1954, Rott 1990):
Reynolds numberRe= ud ν ,
for pipe flow: uaveraged velocity,dpipe diameter,ν kinematic viscosity. The Reynolds number Re is the most important parameter in fluid dynamics. For low Re flows are laminar, for increasingRethey become turbulent at a critical value. What may look rather simple at a first glance, was a breakthrough in fluid dynamics. The Reynolds similarity principle means that flows on e.g completely different length scales are identical to each other, ifReis the same. This is the basis on which large-scale flows can in principle all be studied in a laboratory experiment.
However, Reynolds realized also that perturbations, e.g. waves in the tank feeding the pipe, changed the criticalRefor the onset of turbulence.
Figure 1.2: Reynolds pipe flow experiment with his assistant. The pipe is made of glass with a diameter of 25mm±0.78mm and a length of 1.5m and is horizontally placed inside a tank of water. The water from the tank is entering the pipe through a trumpet mouth made of wood. The height of the water in the tank is measured (see instrument to the right side of the assistant) to obtain the velocity of the flow. A vertically mounted iron tube connects the end of the glass pipe with a valve at ground level. A long lever is connected to this valve that reaches up to the platform, so that the flow rate can be easily controlled. For the flow visualization another tube connects a reservoir of ink (placed on top of the water tank) with the centerline of the trumpet mouth. The figure is a public domain and taken from Reynolds (1883).
Figure 1.3: Visualization of the onset of turbulence. Ink is added at the center of the trumped mouth. If the flow is laminar, the ink stays confined to the center of the pipe (top picture). The onset of turbulence is not directly at the entrance but some distance downstream. It can be seen by the mixing of the ink with the surrounding water (bottom picture). The figure is a public domain and taken from Reynolds (1883).
‘This showed that the steady motion was unstable for large disturbances long before the critical velocity was reached . [. . . ] But the fact that in some conditions it [the laminar flow] will break down for a large disturbance, while it is stable for a smaller disturbance shows that there is a certain residual stability so long as the disturbances do not exceed a given amount’ (Reynolds 1883).
What Reynolds was guessing here, is that pipe flow is linearly stable to infinitesimal perturbations and only finite amplitude perturbations can trigger the transition to turbu-lence. The linear stability was first supported only 100 years later (Salwenet al. 1980, Drazin & Reid 2004). So far it has been proven to be true at least up to Re = 107 in computations (Meseguer & Trefethen 2003) and up to 105 in experiments (Pfenninger 1961). The linear stability of the parabolic velocity profile of pipe flow is the reason, why transition to turbulence is still puzzling researchers nowadays.
But after many experiments Reynolds found a way in which the critical Re for the onset of turbulence could be determined:
‘it became clear to me that if in a tube of sufficient length the water were at first admitted in a high state of disturbance, then as the water proceeded along the tube the disturbance would settle down into a steady condition, which condition would be one of eddies or steady motion, according to whether the velocity was above or below what may be called the real critical value’
(Reynolds 1883).
Reynolds continued the search for this criticalRefor his entire life, even though
‘at first sight such experiments may appear to be simple enough, yet when one began to consider actual ways and means, so many uncertainties and difficulties presented themselves that the necessary courage for undertaking them was only acquired after two years’ further study of the hydrodynamical aspect of the subject by the light thrown upon it by the previous experiment’
(Reynolds 1883).
Figure 1.4: Localization of turbulence. At lowRethe flow is spatio-temporally intermit-tent, consisting of an irregular sequence of laminar and turbulent patches in the stream-wise direction. Reynolds called the turbulent patches ‘flashes of turbulence’, nowadays there commonly referred to as ‘puffs’. The figure is a public domain and taken from Reynolds (1883).
His last estimates were aboutRe=1900, 2000, but he could never get the answer. This is nowadays known as the ‘Reynolds-problem’. At these Re turbulence is intermittent, appearing in form of streamwise localized patches that are swept downstream with the surrounding laminar flow as it is illustrated in Fig. 1.4.