Notes on Computational Fluid Dynamics: General Principles

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2.24 Summary of tensor algebra

Below is tensor algebra applied to Cartesian ( $\text{[math]}$ , $\text{[math]}$ , $\text{[math]}$ ) co-ordinates using: scalar $\text{[math]}$ ; vectors $\text{[math]}$ , $\text{[math]}$ ; tensors $\text{[math]}$ , $\text{[math]}$ , $\text{[math]}$ .

Products

Inner product of two vectors, Sec. 2.3

$s = a b = axbx + ayby + azbz \relax \special {t4ht=$
Outer product of two vectors, Sec. 2.8

$0 1 axbx axby axbz T = ab = B@ aybx ayby aybz CA azbx azby azbz \relax \special {t4ht=$
Inner product of vector and tensor, Sec. 2.6

$0 a T + a T + a T 1 B x xx y yx z zx C b = a T = @ axTxy + ayTyy + azTzy A axTxz + ayTyz + azTzz \relax \special {t4ht=$
Inner product of two tensors, Sec. 2.8
$pict\relax \special {t4ht=$
Double inner product of two tensors, Sec. 2.17

$s = T S = T S + T S + T S + xx xx xy xy xz xz TyxSyx + TyySyy + TyzSyz + TzxSzx + TzySzy + TzzSzz \relax \special {t4ht=$
Cross product of two vectors, ﬁrst used in Sec. 3.3 , produces a vector with components

$a b = (aybz azby;azbx axbz;axby aybx) \relax \special {t4ht=$ (2.70)

Tensors and operations

Transpose of a tensor, Sec. 2.7

$0 1 0 1 B Txx Txy Txz C T B Txx Tyx Tzx C T = @ Tyx Tyy Tyz A T = @ Txy Tyy Tzy A Tzx Tzy Tzz Txz Tyz Tzz \relax \special {t4ht=$
Symmetric and skew tensors, Sec. 2.7

$1 1 T = -(T + TT) + -(T TT) = sym T + skw T 2|------------{z------------} |2------------{z------------} symmetric skew \relax \special {t4ht=$
Trace of a tensor, Sec. 2.10

$tr(T) = T + T + T xx yy zz \relax \special {t4ht=$
Identity tensor, Sec. 2.8
$0 1 B 1 0 0 C I = @ 0 1 0 A 0 0 1 I T T I T T sI str(T) \relax \special {t4ht=$
Deviatoric and spherical tensors, Sec. 2.10

$1 1 T = T 3-(tr T)I + 3 (trT) I = devT + sph T |---------------{z---------------} |--------{z--------} deviatoric spherical \relax \special {t4ht=$

Notes on CFD: General Principles - 2.24 Summary of tensor algebra