Turbulence

So far, a parcel of tracer has been considered, initially stationary and then examined whilst moving in laminar flow. The nearshore zone is subject to a combination of wave and longshore current effects. The flows within these areas are turbulent. Thus in this section some features of turbulent flow will be reviewed.

In open channel or pipe flows, in the vicinity of boundaries, as flow velocities increase, the velocity fluctuations change from a relatively simple form of regular sinusoidal movements to highly irregular movements which lead to the formation of large scale flow structures within the flow, or eddies. The eddies are formed in the regions of high velocity gradients, notably in the vicinity of boundaries. This leads to a diverse range of size and frequency of eddy motions throughout the flow.

The exact formation of how a small velocity fluctuation can produce such highly irregular turbulent flow structure of many times the magnitude of the original movement is unclear and is still the subject of current research. Csanady (1973) noted that certain larger eddies increased in magnitude with time and concluded that these larger eddies combined their energy with smaller eddies, which increased the resultant eddy. These large parcels of fluid movement then in-turn displace other fluid which inevitably has to fill the space vacated by the moving eddy. The movement of fluid into the space vacated by the moving eddy then generates additional smaller eddies which increase by combining together and the process

20 continues until the size of the smallest eddy does not possess enough energy to combine with another eddy, therefore ensuring stability of the velocity movement. Rutherford (1994) referred to previous studies of Fischer et al. (1979) and Chatwin & Allen (1985) who in turbulent river flows, suggested the smallest size of eddy to be the order of ₁₀4_{to 10}3_{m. Thus, so far it has been deduced that turbulence in} pipe flow or open channel flow, is generated in the vicinity of boundaries. Thus it is expected that, for example in the vertical direction, the eddies cannot grow indefinitely, but are restricted in size to the depth of flow. Without going into further detail within this section, it is noted that turbulent flows are characterised by a diverse and complex number of eddies, of varying size and shape, and moving with differing velocities and direction.

Due to the complexities of turbulent flows, Reynolds defined the turbulent properties of the flow by adopting a statistical approach based on the small scale random particle fluctuations caused by the large scale eddy motion within the flow. Reynolds sought to isolate the velocity fluctuations from advection and derived the Reynolds equations of motion, more commonly known as the Reynolds’ Rules of Averaging, given by Holly (1985):

u u

u    (2.18)

where the u is an average of the longitudinal velocity over a representative period of time and u is the instantaneous local deviation from the mean value. The turbulent intensity, or degree of turbulence within a flow at particular elevation, is given by:

) (u 2

um   (2.19)

um is sometimes referred to as a ‘characteristic turbulent velocity’, which indicates

the behaviour of the eddy movements (Csanady, 1973). Reynolds’ deduced that the turbulent velocity fluctuations generated forces or stresses within the flow, and proposed that the stresses could be described mathematically. The basic principle governing Reynolds’ theoretical treatment was that momentum is conserved during the movement of fluid. For simplicity, consider the case of two-dimensional flow where it is assumed that only movements in the x and z direction occur. Figure 2.4

21 shows the movement of a small element of fluid passing through a small horizontal surface of area A , whose dimensions are xy.

Figure 2.4 – Reynolds’ stress eddy model, adapted from Chadwick & Morfet (1986) During a small time interval, t it is assumed that the mass of fluid flowing through the surface in the z-direction is given by:

t A w 

 _(2.20)

By adopting the Rules of Averaging, in the x-direction, the mass of fluid has the instantaneous horizontal velocity component of u u. By definition the momentum

)

(M of the mass of fluid is therefore given by:

) )(

( w A t u u

M     

  (2.21)

The rate of change of momentum of the fluid over the small time interval, t is therefore given by:

) ( ) ( w Au wu A t M _{    }   _{ } (2.22)

Over the small time interval, the average rate of transport of momentum is related to the time averaged velocities of the turbulent fluid motion. Thus, by definition, although the mean of the velocity fluctuation in the z direction (w) is zero, the mean of the product (wu) generally yields non-zero results. From Newton’s second law, the rate of change of momentum equals a force, and stress is the ratio of force over area, hence, the average stress over the small time interval is given by;

22 u

w t   

 (2.23)

where t is denoted by turbulent shear stress and is more commonly known as a

Reynolds’ stress.