APPENDIX A
Statistics
A.1 Reynolds operator
In fluid dynamics a Reynolds operatorhi is an averaging operator that:
• satisfies the linearity property;
• satisfies the following averaging properties:
hx + yi = hxi + hyi haxi = ahxi
hhxiyi = hxihyi, which implies hhxii = hXi = X = hxi.
being x and y two random variables, and a an arbitrary constant.
• is a commuting operator in space and time:
∂x
∂t
= ∂hxi
∂t
∂x
∂r
= ∂hxi
∂r
Z
x(r, t) dr dt
= Z
hx(r, t)i dr dt.
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APPENDIX A. STATISTICS 81
A.2 Variance (second central moment)
Considering the properties of Reynolds operators, it is var[x] =hx02i = h(x − X)2i = hx2− 2xX + X2i =
=hx2i − 2hxXi + hX2i = hx2i − 2hxiX + X2 =
=hx2i − 2X2+ X2 =
=hx2i − X2
A.3 Third central moment
Considering the properties of Reynolds operators, it is
hx03i = h(x − X)3i = hx3− 3x2X + 3xX2− X3i =
=hx3i − 3hx2Xi + 3hxX2i − hX3i =
=hx3i − 3hx2iX + 3hxiX2− X3 =
=hx3i − 3hx2iX + 3XX2− X3 =
=hx3i − 3hx2iX + 2X3
A.4 Fourth central moment
Considering the properties of Reynolds operators, it is
hx04i = h(x − X)4i = hx4− 4x3X + 6x2X2− 4xX3+ X4i =
=hx4i − 4hx3Xi + 6hx2X2i − 4hxX3i + hX4i =
=hx4i − 4hx3iX + 6hx2iX2− 4hxiX3+ X4 =
=hx4i − 4hx3iX + 6hx2iX2− 4XX3+ X4 =
=hx4i − 4hx3iX + 6hx2iX2− 3X4
