Curvilinear Coordinates Outline 1 Orthogonal curvilinear coordinate systems 2 Differential operators in orthogonal curvilinear coordinate systems 3 Derivatives of the unit vectors in orthogonal curvilinear coordinate systems 4 Incompressible N S equations in orthogonal curvilinear coordinate systems 5 Example Incompressible N S equations in cylindrical polar systems The governing equations were derived using the most basic coordinate system i e Cartesian coordinates x x i y j zk f f f grad f f i j k x y z F F F div F F 1 2 3 x y z i j k curl f F x y z F1 F2 F3 Laplacian 2 f 2 f 2 f 2 f 2 2 x2 y z Example incompressible flow equations V 0 DV r p gz m 2 V Dt V r V V p gz m 2 V t V 1 r V V V p gz m V t 2 V V 0 in the above equation but retained to keep the complete vector identity 2 for V in equation However once the equations are expressed in vector invariant form as above they can be transformed into any convenient coordinate system through the use of appropriate definitions 2 for the and Frequently alternative coordinate systems are desirable which either exploit certain features of the flow at hand or facilitate numerical procedures The most general coordinate system for fluid flow problems are nonorthogonal curvilinear coordinates A special case of these are orthogonal curvilinear coordinates Here we shall derive the appropriate relations for the latter using vector technique It should be recognized that the derivation can also be accomplished using tensor analysis 1 Orthogonal curvilinear coordinate systems Suppose that the Cartesian coordinates x y z are expressed in terms of the new coordinates x1 x2 x3 by the equations x x x1 x2 x3 y y x1 x2 x3 z z x1 x2 x3 where it is assumed that the correspondence is unique and that the inverse mapping exists For example circular cylindrical coordinates x r cos q y r sin q z z i e at any point P x1 curve is a straight line x2 curve is a circle and the x3 curve is a straight line The position vector of a point P in space is R x i y j zk R r cos q i r sin q j z k for cylindrical coordinates By definition a vector tangent to the x1 curve is given by R x xx i y x j z x k Subscript denotes partial differentiation 1 1 1 1 So that the unit vectors tangent to the xi curve are e 1 R x1 h1 e 2 R x2 h2 e 3 R x3 h3 Where hi R x are called the metric coefficients or scale factors hr 1 hq r hz 1 for cylindrical coordinates 1 The arc length along a curve in any direction is given by ds 2 dR dR h12 dx12 h22 dx22 h32 dx32 Since dR R xi dxi hi dxi e i and R xi hi e i 0 e j and since the xi are orthogonal e i 1 i j i j An element of volume is given by the triple product d h1dx1e 1 h2 dx2e 2 h3dx3e 3 h1h2 h3dx1dx2 dx3 Where since the xi are orthogonal e 1 e 2 e 3 Finally on the surface x1 constant the vector element of surface area is given by ds1 h2 dx2e 2 h3dx3e 3 e 1h2 h3 dx2 dx3 With similar results for x2 and x3 constant ds 2 e 2 h3 h1dx3 dx1 ds3 e 3 h1h2 dx1dx2 2 Differential operators in orthogonal curvilinear coordinate systems With the above in hand we now proceed to obtain the desired vector operators 1 f 1 f 1 f 2 1 Gradient f h x e 1 h x e 2 h x e 3 1 1 2 2 3 3 dR f xi dxi By definition df f If we temporarily write f l 1e 1 l 2e 2 l 3e 3 Then by comparison df f xi dxi l i hi dxi 1 f hi xi 1 1 1 e 1 e 2 e 3 h1 x1 h2 x2 h3 x3 1 e r e q e z for cylindrical coordinates r r q z li e i hi 1 f li hi xi Note xi So that by definition curl grad f 0 xi Also e i hi 0 e 1 e 2 e 3 x2 h2 h3 h2 h3 x3 f g So that by definition 0 e 1 e 2 e 3 0 h h h h h1h2 2 3 3 1 F 2 2 Divergence 1 h1h2 h3 h2 h3 F1 h3 h1F2 h1h2 F3 x1 x2 x3 F F1e 1 F2e 2 F3e 3 e u u j F1e 1 h2 h3 F1 1 using j u j h2 h3 e e 1 e 2 e 3 1 h2 h3 F1 using 0 h2 h3 h1h2 h2 h3 h3h1 1 h2 h3 F1 h1h2 h3 x1 Treating the other terms in a similar manner results in 1 F h2 h3 F1 h3 h1F2 h1h2 F3 h1h2 h3 x1 x2 x3 1 F rF1 F2 rF3 r r q z 1 1 rF1 F2 F3 for cylindrical coordinates r r r q z h1e 1 1 2 3 Curl F h1h2 h3 x1 F h2e 2 x2 h3e 3 x3 h1F1 h2 F2 h3 F3 F1e 1 F2e 2 F3e 3 e h1 F1e 1 h1F1 1 e 1 h1 F1 using h1 j u j u and u j e i 0 hi e 1 1 h1 F1 1 h1F1 1 h1 F1 e 1 e 2 e 3 h1 h1 x1 h2 x2 h3 x3 e 3 e 2 h1F1 h1F1 h1h2 x2 h3h1 x3 F 1 h2e 2 h3e 3 h1F1 h1h2 h3 x3 x2 h1e 1 h2e 2 h3e 3 1 h1h2 h3 x1 x2 x3 h1 F1 h2 F2 h3 F3 e r re q e z 1 F for cylindrical coordinates r r q z F1 rF2 F3 2 2 4 Laplacian acting on a scalar f 1 h1h2 h3 h2 h3 h3 h1 h1h2 x1 h1 x1 x2 h2 x2 x3 h3 x3 2 h2 h3 h3h1 h1h2 x1 h1 x1 x2 h2 x2 x3 h3 x3 1 1 2 r r r r r q r q z z 1 1 1 1 r r r r r r q r q r z z 1 h1h2 h3 2 F F 2 5 Laplacian acting on a vector Using f F and F F 1 f 1 f 1 f e 1 e 2 e 3 h1 x1 h2 x2 h3 x3 1 h1h2 h3 h2 h3 F1 h3 h1F2 h1h2 F3 x1 x2 x3 1 1 e 1 h2 h3 F1 h3h1F2 h1h2 F3 h1 x1 h1h2 h3 x1 x2 x3 1 1 e 2 h2 h3 F1 h3h1F2 h1h2 F3 h2 x2 h1h2 h3 x1 x2 x3 1 1 e 3 h2 h3 F1 h3 h1 F2 h1h2 F3 h3 x3 h1h2 h3 x1 …
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