INVISCID FLOW Week 1

2017 Spring Inviscid Flow2017 Spring

INVISCID FLOWWeek 1

Prof. Hyungmin ParkMultiphase Flow and Flow Visualization Lab.

Department of Mechanical and Aerospace EngineeringSeoul National University

2017 Spring Inviscid Flow

Course Introductiono Textbook

– Main• Lecture Slide – please check “eTL” before the class• Fundamental Mechanics of Fluids (I. G. Currie), 3rd Ed., Marcel

Dekker, Inc. (2003)– References

• Inviscid Incompressible Flow (J. S. Marshall)• Physical Fluid Dynamics (D. J. Tritton),• Vectors, Tensors and the Basic Equations of Fluid Dynamics (R.

Aris)• An Album of Fluid Motion (M. Van Dyke)• Papers from archival journals• ETC.

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Tensoro What is tensor?

– Difficult to define with a simple word, but it refers to objects that have multiple indices.• e.g., a “scalar” is a tensor of 0th order, a “vector” is a tensor of 1st

order– Adopted to describe physical phenomena irrespective of coordinate

systems– To avoid tedious work of writing long equations with various terms, we

have made some rules with tensor– Offers a more compact form of governing equations in various

disciplines, e.g., fluid mechanics, solid mechanics, electrical engineering……

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Tensoro Index notation (in Cartesian coordinates)

– Einstein’s summation convention

• Free index: If a given index appears only once in each term of a tensor equation, it is a free index and the equation holds for all possible values of that index.

Ex) i = 1, 2, 3 xi = (x1, x2, x3)

• Dummy index: If an index appears twice in any term, it is understood that a summation is to be made over all possible values of that index.

Ex) i = 1, 2, 3 aibi = a1b1 + a2b2 + a3b3

aibicj = a1b1cj + a2b2cj + a3b3cj

• No index may appear more than twice in any term.

– Dummy index may appear only twice in any given term. cf) aibici

– Same free indices must appear in every term in an equation.

cf) ai = 3 + bi

4


Tensoro A tensor of rank r is a quantity having nr components in n-dimensional

space. The components of a tensor quantity expressed in two different coordinate system are related as follows:

o Rank 0 tensor in 3-dimensional space à a scalar having 1 component

o Rank 1 tensor in 3-dimensional space à a vector having 3 components

o Rank 2 tensor is called a dyadicà represented by a matrix

– r =2 and n =3 à 3 x 3 matrix (9 components)

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ijk m is jt ku mv stu vT C C C C T××× ×××¢ = × × ×

Directional cosine between the axes of two different coordinate systems.

11 12 13

21 22 23

31 32 33

ij

a a aA a a a

a a a

æ öç ÷= ç ÷ç ÷è ø(i,j = 1,2,3)


Tensoro When do we use the tensor?

– Governing equations such as Navier-Stokes Equation

– Turbulence modeling

– Coordinate transformation– Grid generation in CFD– Matrix calculation– …….

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21i i ij

j i j j

u u p uut x x x x

nr

¶ ¶ ¶ ¶+ = - +

¶ ¶ ¶ ¶ ¶

23

( )jii j ij T

j i

uuu u kx x

d n¶¶

= - +¶ ¶


Tensoro Tensor Algebra

– Addition: Two tensors of equal rank may be added to yield a third

tensor of the same rank.

– Multiplication: If tensor A has rank ‘a’ and tensor B has rank ‘b’, the

multiplication of these two tensors yields a third one of rank ‘c’.

– Contraction: If any two indices of a tensor of rank r ³2 are set equal, a

tensor of rank r-2 is obtained

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.... .... ....ij k ij k ij kC A B= +

c = a + b.... .... .... ....ij krs t ij k rs tC A B=

ij i jC AB=

1 1 2 2 3 3ii i iC AB AB A B A B= = + +Tensor C of rank 2

If i=j, tensor of rank 0 (scalar) is

obtained


Tensoro Tensor Algebra

– Symmetry: If the tensor A has the property below, then it is said to be symmetric in the indices j and k. As a consequence of the relation above, the tensor has nr-1(n+1)/2 independent components in n-dimensional space. Here, r is the rank of a Tensor.

– Anti-symmetry: If the tensor A has the property below, then it is said to be anti-symmetric in the indices j and k. As a consequence of the relation above, the tensor has nr-1(n-1)/2 independent components in n-dimensional space. Here, r is the rank of a Tensor.

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... ... ... ... ... ...i j k l i k j lA A=

... ... ... ... ... ...i j k l i k j lA A= -


Tensoro Tensor Operation

– Gradient: the gradient of tensor R (rank of r) yields a tensor T (rank of r+1).

– Divergence: the gradient of tensor R (rank of r) yields a tensor T (rank of r-1).

– Curl: If R is a tensor of rank r, the curl operation will produce an anti-symmetric tensor of rank (r+1).

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........

ij kij kl

t

RT

x¶

=¶

.... ........ ....

i jkl mi jl m

k

RT

x¶

=¶

.... .... .... ........ ....

i j k i l ki j kl

l j

R RTx x

¶ ¶= -

¶ ¶


Tensoro Identity tensor, I: components dij are called the Kronecker delta.

– Kronecker delta: dij = 0 (i ¹ j) or 1 (i = j)

– Multiplication of the components of any vector by yields a change in the index.

• aidij = a1d1j+a2d2j+a3d3j = a3d3j = aj (if j = 3)

– In a Cartesian coordinate system, if X denotes the position vector with components xi, then

o Permutation tensor: components are defined as eijk which is 1 if (i, j, k) are cyclic, -1 if (i, j, k) are anti-cyclic, and 0 if any of (i, j, k) are the same.

o Useful identity from Kronecker delta and permutation tensor:

eijkeimn = djmdkn - djndkm

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iij

j

xx

d¶=

¶


Tensoro Isotropic Tensors

– An isotropic tensor is one whose components are invariant with respect to all possible rotations of the coordinate system. That is, there are no preferred directions, and the quantity represented by the tensor is a function of position only and not of orientation.

– Isotropic tensors of Rank 0: all tensors of rank 0 (i.e., scalars) are isotropic.

– Isotropic tensors of Rank 1: there are no isotropic tensors of rank 1. That is, vectors are not isotropic, since there are preferred directions.

– Isotropic tensors of Rank 2: the only isotropic tensors of rank 2 are of the form adij, where a is a scalar and dij is the Kronecker delta.

– Isotropic tensors of Rank 3: the isotropic tensors of rank 3 are of the form aeijk, where a is a scalar and eijk is a permutation tensor.

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Tensoro Del Operator (Ñ)

– Definition

– Divergence of a vector u

– Curl of u

– Laplacian of a scalar f

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ii

ex¶

Ñ º¶

i

i

uux¶

Ñ × =¶

kijk i

j

uu ex

e ¶Ñ´ =

¶

22

i ix xff f ¶

Ñ =Ñ×Ñ =¶ ¶


Tensoro Symmetric and Skew-symmetric Tensor

– A second order tenor A is symmetric if A = AT, or Aij = Aji

– It is said to be skew-symmetric if A = -AT, or Aij = -Aji

– The diagonal components of a skew-symmetric tensor are zero– e.g., Kronecker delta (dij) is symmetric

Permutation tensor (eijk) is skew-symmetric in any two of indices

– If A is symmetric and B is skew-symmetric tensors, respectively, the scalar product of A and B is zero.

AijBji = AjiBji = -AjiBij = -AijBji, \ AijBji = 0

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Tensoro Useful Vector Identities – can be proved with tensor notation

– Ñ ´ Ñ f = 0– Ñ × (Ñ ´ a) = 0– Ñ2a = Ñ(Ñ × a) - Ñ ´ (Ñ ´ a)– Ñ × (fa) = f(Ñ × a) + a × Ñf– Ñ ´ (fa) = f(Ñ ´ a) + (Ñf) ´ a– Ñ(a × b) = (a × Ñ)b + (b × Ñ)a + a ´ (Ñ ´ b) + b ´ (Ñ ´ a)– Ñ × (a ´ b) = b × (Ñ ´ a) - a × (Ñ ´ b)– Ñ ´ (a ´ b) = a(Ñ × b) - b(Ñ × a) + (b × Ñ)a - (a × Ñ)b– a ´ (b ´ c) = (a × c)b - (a × b)c

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Tensoro Example 1

– Ñ × (Ñ ´ a) = 0

15

2

0( )

kijk i

j

kijk

i j

aa ex

aax x

e

e

¶Ñ´ =

¶

¶\Ñ× Ñ´ = =

¶ ¶

Skew-symmetric

Symmetric in i and j


Tensoro Example 2

– Ñ × (a ´ b) = b × (Ñ ´ a) - a × (Ñ ´ b)

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( ) ( )

( ) ( )

ijk j k i

j kijk j k ijk k ijk j

i i i

j jk kkij k jik j k kij j jik

i i i i

a b a b ea ba b a b b a

x x xa ab bb a b ax x x x

b a a b

e

e e e

e e e e

´ =

¶¶ ¶Ñ × ´ = = +

¶ ¶ ¶¶ ¶¶ ¶

= - = -¶ ¶ ¶ ¶

= × Ñ´ - × Ñ´


Tensoro Example 3

– a ´ (b ´ c) = (a × c)b - (a × b)c

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( ) ( )( )

( )( )

( ) ( )

klm l m k

ijk j klm l m i ijk klm j l m i

kij klm j l m i il jm im lj j l m i

il jm j l m im lj j l m i

j i j j j i i

j j i i j j i i

b c b c ea b c a b c e a b c e

a b c e a b c ea b c a b c e

a b c a b c ea c b e a b c ea c b a b c

ee e e e

e e d d d dd d d d

´ =´ ´ = =

= = -

= -

= -

= -

= × - ×

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INVISCID FLOW Week 1