C,3 • FREQUENCY DISTRIBUTIONS. STATISTICAL AVERAGES 



F(u[, x[; u[' , x'{\ t). The correlation of the x components of the velocities 

 at these points is then given by 



u{P'MP") = jj F{u', v'w'; x', y', z'; u", v" , w"; x", y", z", t) 



u'u"d{u', v', w')d{u", v", w") (3-2) 



Further generaUzation of joint probability distributions involves quanti- 

 ties observed at more than two points, and quantities other than velocity 

 fluctuations. 



Some experimental information is available regarding the distribution 

 function F{ui, Xi, t). The Gaussian distribution has been found, in many 

 cases, to be a fairly good approximation for each component (Fig. C,3a).^ 

 In the isotropic case, i.e. where the statistical properties of the motion 



^-x^ 



.^ 



_X' 



V 



X Measurements 

 (x/M= 16, UM/v = 9600) 



Normal distribution 



P(u,) 



'^--, 



X- 



U] fluctuation 



Fig. C,3a. Probability density function of the velocity component ui in the direction 

 of the stream for the turbulence generated by a square-mesh grid in a wind tunnel 

 (after [1]). 



are essentially independent of direction, the Maxwellian distribution of 

 velocity holds approximately [9,10]. 



There is also some indication that the joint probabihty distribution 

 at two points in an isotropic field is approximately jointly Gaussian. How- 

 ever, this is known to be not accurate. To get a quantitative assessment 

 of the departure from j oint Gaussian distribution, one may introduce the 

 quantities 



and 



Sir) = 



F(r) = 



(3-3) 



\{u' - uy\ 



' In this and the following figures, M denotes the width of the mesh of grid, x 

 denotes the distance of the observation point from the grid, U denotes the velocity of 

 air, and v denotes the kinematic viscosity coefficient. 



< 199 ) 



