Differential Approach DIFFERENTIAL ANALYSIS OF FLUID FLOW • Finite control volume approach is very practical and useful, since it does not generally require a detailed knowledge of the pressure and velocity variations within the control volume • Problems could be solved without a detailed knowledge of the flow field • Unfortunately, there are many situations that arise in which details of the flow are important and the finite control volume approach will not yield the desired information • How the velocity varies over the cross section of a pipe, how the pressure and shear stress vary along the surface of an airplane wing •In these circumstances we need to develop relationships that apply at a point, or at least in a very small region infinitesimal volume within a given flow field. This approach - DIFFERENTIAL ANALYSIS • DIFFERENTIAL ANALYSIS PROVIDES VERY DETAILED KNOWLEDGE OF A FLOW FIELD Flow domain Control volume Flow out Flow out Flow in Flow in Flow out Flow out )& F )& F Control volume analysis Differential analysis Interior of the CV is All the details of the flow are solved at every point within the flow domain BLACK BOX LINEAR MOTION AND DEFORMATION Element at t0 Element at t0+δt + = General motion Translation + Linear deformation + Rotation Angular deformation TRANSLATION O’ v O u vδt uδt If all points in the element have the same velocity which is only true if there are no velocity gradients, then the element will simply TRANSLATE from one position to another. LINEAR DEFORMATION wu Gx u C wx B u δy C C’ δy u u O B δx A wu Gx wx O δx A A’ § wu · ¨ wx G x ¸ G t © ¹ Because of the presence of velocity gradients, the element will generally be deformed and rotated as it moves. For example, wu consider the effect of a single velocity gradient On a small cube having sides G x , G y and G z wx x component of velocity of O and B = u wu Gx x component of velocity of A and C = u wx This difference in the velocity causes a “STRETCHING” of the volume element by a volume § wu · ¨¨ G x ¸¸ G y G z G t ©wx ¹ Rate at which the volume GV is changing per unit volume due wu the gradient wx 1 d GV GV dt ª wu wt º wx » Lim « » wt G t o0 « ¬ ¼ wu wx wv ww & If the velocity gradients are also present wy wz 1 d GV GV dt wu wv ww wx wy wz This rate of change of volume per unit volume is called the VOLUMETRIC DILATION RATE Volume of the fluid may change as the element moves from one location to another in the flow field Incompressible fluid – volumetric dilation rate = zero Change in volume element = zero; fluid density = constant (The element mass is conserved) Variations in the velocity in the direction of velocity cause LINEAR DEFORMATION wu wv ww & , wx wy wz Linear deformation of the element does not change the shape of the element Cross derivates cause the element to ROTATE and undergo ANGULAR DEFORMATION wu wv , wy wx Angular deformation of the element changes the shape of the element ANGULAR MOTION AND DEFORMATION B u wu Gy wy § wu · ¨ G y ¸G t © wy ¹ C C B’ B GE δy δy v wv v Gx wx u O A’ δx A § wv · x G ¨ wx ¸G t © ¹ GD O δx A Consider x-y plane. In a short time interval Gt line segment OA and OB will rotate through angles GD and GE to the new positions OA’ and OB’ Angular velocity of OA, ZOA Z oA GD Lim G t o0 G t For small angles wv Tan GD | GD wv wx wx G xGt Gx - positive § wu · ¨ wy ¸ © ¹ § wu · G y G t ¨ wy ¸ ¹ © G y - positive Z oA wv wx Z oA - counterclockwise Z oB TanGE | GE wv Gt wx § wv · G t ¨ ¸ wx Lim ¨ ¸¸ G t ¨ G t o0 © ¹ GE Lim G t o0 G t wu Gt wy Z oB - clockwise Z oB § § wu · · ¨ ¨ wy ¸ G t ¸ © ¹ ¨ ¸ Lim ¸ Gt G t o 0 ¨¨ ¸ © ¹ wu wy Rotation Zz of the element about the z-axis is defined as the average of the angular velocities ZoA and ZoB of the two mutually perpendicular lines OA and OB. Thus, if counterclockwise rotation is considered positive, it follows that Zz wu· 1§ wv ¨¨ ¸¸ 2©w x w y¹ Rotation Zx of the element about the x-axis Zx 1 § ww wv · ¨¨ ¸¸ 2© w y wz¹ Rotation Zyof the element about the y-axis Zy Z 1§wu ww· ¨¨ ¸¸ wx¹ 2©wz Z x î Z y ĵ Z z k̂ Z 1 curl V 2 1 uV 2 Vorticity : is defined as the vector that is twice the rotation vector : 2Z uV Fluid element will rotate about the z axis as an undeformed block wu wv (ie., ZoA = - ZoB ) only when Otherwise, the wy wx rotation will be associated with an angular deformation wv wx uV wu wy 0 Rotation around the z axis is zero. Rotation and vorticity are zero; FLOW FIELD IS IRROTATIONAL In addition to rotation associated with derivatives w u wy & wv wx These derivatives can cause the fluid element to undergo an angular deformation which results in change of shape Change in the original right angle formed by the lines OA and OB is SHEARING STRAIN GJ GJ = GD + GE GJ is positive if the original right angle is decreasing Rate of Shearing Strain or Rate of Angular Deformation J § GJ · Lim ¨ ¸ Gt 0 © Gt ¹ ª wv § wu · G G t ¨ wy ¸ « wx © ¹ « Lim Gt Gt 0 « «¬ º t» » » »¼ wu wv wx wy J wu wv wy wx Rate of angular deformation is related to a corresponding shearing stress which causes the fluid element to change in shape wu wy wv wx Rate of angular deformation is zero; Rotation Element is simply rotating as an undeformed block Volume = V2= V1 Time = t2 Incompressible flow field Fluid elements may translate, distort, and rotate but do not grow or shrink in volume Time = t1 Volume = V1 Compressible flow field (a) Time = t1 Volume = V1 Fluid elements may grow or shrink in volume as they translate, distort or rotate Time = t2 (b) Volume = V2 CONSERVATION OF MASS OR CONTINUITY EQUATION DBsys Dt w Gt ³ cv UbdV w ³ U dV ³ UV x n̂dA G t cv cs ³ UbV x n̂dA x1 z1 cs y1 0 y dx dz x dy z Time rate of change of the mass of the coincident system Time rate of change of the mass of the contents of the coincident control volume Net rate of flow of mass through the control surface w wU GxGyGz ³ U dV wt G t cv ³ UV x n̂dA w Uv Uv G x G z GxG y G z wy cs w Uw Uw G x G y GxG y G z wz Uu G y G z Gy K iG z Uw G x G y j Gx w Uu Uu G y G z GxG y G z wx Uv G x G z w ³ U dV ³ UV x n̂dA G t cv cs 0 wU GxGyGz Uu G y G z Uv G x G z Uw G x G y Uu G y G z wt w Uu w Uv w Uw GxG y G z Uv G x G z GxG y G z Uw G x G y GxG y G z wy wz wx w Uu w Uv w Uw wU GxGyGz GxG y G z GxG y G z GxG y G z wx wy wz wt w Uv w Uw wU w Uu wt wx wy wz 0 0 0 wU w Uu w Uv wt wx wy w Uw wz 0 ww wv wU wU wU wu wU U v U w u U wy wz wz wx wx wy wt ª wu wv ww º wU wU wU wU u v w U« » wt wx wy wz ¬w x w y w z ¼ > @ DU U .V̂ Dt 0 0 0 Determine the form of the z-component, w, required to satisfy the continuity equation. The velocity components for a certain incompressible, steady flow field are as follows. u x2 y2 z2 v xy yz z w ? wu wv ww wx w y wz 0 ww 2x x z wz 0 ww wz 3x z 2 2 w z 3 xz C 2 w z 3 xz f x , y 2 CONSERVATION OF MOMENTUM w ³ UVd V ³ VUV x n̂dA G t cv cs RATE AT RATE OF INCREASE OF - WHICH xMOMENTUM x-MOMENTUM ENTERS w ³ UVd V G t cv ¦ Fcontents of control volume RATE AT + WHICH xMOMENTUM LEAVES = w Uu GxGyGz wt SURFACE FORCES BODY FORCES • NORMAL STRESSES • GRAVITY FORCES • SHEAR STRESSES • CORIOLIS FORCES • PRESSURE • CENTRIFUGAL FORCES SUM OF THE X-COMP FORCES APPLIED TO FLUID IN CV ³ VUV x n̂dA cs u Uv G y G z w Uuv GxG y G z wy u Uw G x G y u UuG y G z iG z u Uw G x G y w Uu 2 u Uu G y G z GxG y G z wx Gy K w Uuw GxG y G z wz j Gx u Uv G x G z w Uu GxGyGz u Uu G y G z u Uv G x G z u Uw G x G y wt w Uu 2 w Uuv u Uu G y G z GxG y G z u Uv G x G z GxG y G z wx w y w Uuw u Uw G y G z GxG y G z LHS wz wUu wUu 2 wUuv wUuw wt wx wy wz § wU w Uu w Uv w Uw u¨¨ wy wy wx © wt LHS GxG y G z § wu · wu wu wu ¸¸ U ¨¨ u v w wx wy wz © wt ¹ Du U Dt LHS GxG y G z · ¸¸ ¹ LHS GxG y G z Vyy Vxx Vxz Vxy Vyz Vyx Vxy Vxz Vxx First subscript denotes the direction of the normal to the plane on which the stress acts Second subscript denotes the direction of the stress Outward normal to the area ABCD – Positive x direction Positive normal stress are tensile stresses – they stretch the material Vxx, Wxy, Wxz are shown in the positive direction wV yx wV zx wP wV xx RHS fx wx wy wz wx GxG y G z V yx Gx Gz wy GxGy Gz V xx Gy Gz V xx Gy Gz Gy K P Gy Gz wV yx iG z j Gx P GyGz wV xx Gx Gy Gz wx wP Gx Gy Gz wx V yx Gx Gz Du U Dt wV yx wV zx wP wV xx fx wz wy wx wx CAUCHY’S EQN V xx wu 2 2P P .V̂ wx 3 V xy wP w ª wu 2 Du P .V̂ U 2P w x w x «¬ wx 3 Dt § wu wv · P ¨¨ ¸¸ © wy wx ¹ V xz § wu ww · P¨ ¸ © wz wx ¹ w ª § wu ww ·º º w ª § wu wv ·º ¨ ¸ » w y « P ¨ wy wx ¸» w z « P ¨ wz wx ¸» f x ¼ ¹¼ ¬ © ¹¼ ¬ © ªw 2u w 2u w 2uº wP Du w ª § wu wv ww P« U » « P ¨¨ 2 2 2 wx Dt wy wz »¼ w x ¬ © wx wy wz «¬ wx · 2 ¸¸ P .V̂ ¹ 3 º » fx ¼ Du U Dt ª w 2u w 2u w 2u º wP w ª P « » P .V̂ « 2 2» wx¬ 3 wx « wx 2 w w y z ¬ ¼ º » fx ¼ Dv U Dt ª w 2v w 2v w 2v º wP w ªP P« .V̂ » « 2 2 2 wy wy wz »¼ w y ¬ 3 «¬ wx º » fy ¼ ª w 2u w 2u w 2u º w ª P Du wP .V̂ U P« » « 2 2 2 Dt wx wz »¼ w x ¬ 3 wy «¬ wx º » fx ¼ Dv U Dt ª w 2v w 2v w 2v º wP w ªP P« .V̂ » « 2 wy wy 2 wz 2 »¼ w y ¬ 3 «¬ wx º » fy ¼ Dw U Dt ª w2w w2w w2w º w ªP wP .V̂ P« » « 2 wz wy 2 wz 2 »¼ w z ¬ 3 «¬ wx º » fz ¼ VISCOUS COMPRESSIBLE FLUID WITH CONSTANT VISCOSITY DV̂ U Dt P P V̂ 2 P 3 .V̂ f VISCOUS COMPRESSIBLE FLUID WITH CONSTANT VISCOSITY DV̂ U Dt P P V̂ 2 P 3 .V̂ f VISCOUS INCOMPRESSIBLE FLUID WITH CONSTANT VISCOSITY DV̂ U Dt P P V̂ f 2 INVISCID INCOMPRESSIBLE FLUID WITH CONSTANT VISCOSITY DV̂ U Dt P f EULER’S EQN ª wu wu wu º wu U« u v w » wy wz ¼ wx ¬ wt Along a stream line wu Uu ds ws Uudu wu Uu ws p wp wz Ug ws ws wp wz ds Ug ds ws ws dp Ugdz u2 U P Ugz 2 2 u gz U 2 wp Ug wx C C Gs gsinT T g Gz Continuity equation X- momentum > @ DU U .V̂ Dt 0 ª w 2u w 2u w 2u º Du w ª P wP U .V̂ P« » « 2 Dt wx wz 2 »¼ w x ¬ 3 wy 2 «¬ wx º » fx ¼ ª w 2v w 2v w 2v º w ªP wP .V̂ P« » « 2 wy wy 2 wz 2 »¼ w y ¬ 3 «¬ wx º » fy ¼ ª w2w w2w w2w º w ªP wP .V̂ P« » « 2 2 2 wz wy wz »¼ w z ¬ 3 «¬ wx Navier – French mathematician Stokes – English Mechanician º » fz ¼ Y- momentum Dv U Dt Z- momentum Dw U Dt FOUR EQUATION AND FOUR UNKNOWNS – U,V,W AND P Mathematically well posed Nonlinear, second order partial differential equations Relation between Stress and Rate of Strain Relation between Stress and Rate of Strain • In elasticity, the relationship between the stress and strain of a solid body within the elastic limit is governed by Hooke’s Law. • The generalised Hooke’s law states that each of the six stress components may be expressed as a linear function of the six components of strain and vice versa • The validity of this assumption has been verified by experiments for continuous, homogenous and isotropic materials. • In a fluid, the physical law connection the stress and rate of strain can also be made by the following simple and reasonable assumptions: a. The stress components may be expressed as a linear function of the rates of strain components b. The relations between stress components and rates of strain components must be invariant to a coordinate transformation consisting of either a rotation or a mirror reflection of axes. c. The stress components must reduce to the hydrostatic pressure ‘p‘ when all the gradients of velocities are zero V xx A1H xx B1H yy C1J xy D1 V yy A2H xx B2H yy C 2J xy D2 V xy A3H xx B3H yy C 3J xy D3 (1) where the A’s, B’s, C’s and D’s are constants to be determined. The assumption (b) requires that the stress-rate of strain relation remains unaltered with respect to a new coordinate system. V xcxc A1H xcxc B1H ycyc C1J xcyc D1 V ycyc A2H xcxc B2H ycyc C 2J xcyc D2 V xcyc A3H xcxc B3H ycyc C 3J xcyc D3 (2) Transformation of stress components y x’ y Y1 y’ C y’ E Y2 V xy V x' x' V x' y' V xx V xy V xx X2 X1 T T o V xcxc V ycyc V xcyc x’ V xy 2 V xx V yy 2 B V yy V xx V yy V xx V yy 2 x V xx V yy 2 V xx V yy 2 D V xy V yy o x cos 2T V xy sin 2T cos 2T V xy sin 2T sin 2T V xy cos 2T (3) Substituting equation (1) into equation (3) V xx A1H xx B1H yy C1J xy D1 V yy A2H xx B2H yy C 2J xy D2 V xy A3H xx B3H yy C 3J xy D3 V xcxc V ycyc V xcyc V xcxc V xx V yy 2 V xx V yy 2 V xx V yy 2 V xx V yy 2 V xx V yy 2 (1) cos 2T V xy sin 2T cos 2T V xy sin 2T (3) sin 2T V xy cos 2T A1H xx B1H yy C1J xy D1 A2H xx B2H yy C 2J xy D2 2 A1H xx B1H yy C1J xy D1 A2H xx B2H yy C 2J xy D2 2 A3H xx B3H yy C 3J xy D3 sin 2T V xcxc A B · §B § A1 · 1 cos 2T 2 1 cos 2T A3 sin 2T ¸ H yy ¨ 1 1 cos 2T 2 1 cos 2T B3 sin 2T ¸ 2 2 © 2 ¹ © 2 ¹ C D · §C · §D J xy ¨ 1 1 cos 2T 2 1 cos 2T C 3 sin 2T ¸ ¨ 1 1 cos 2T 2 1 cos 2T D3 sin 2T ¸ 2 2 ¹ © 2 ¹ © 2 H xx ¨ cos 2T V xcxc A B · §B · § A1 1 cos 2T 2 1 cos 2T A3 sin 2T ¸ H yy ¨ 1 1 cos 2T 2 1 cos 2T B3 sin 2T ¸ 2 2 ¹ © 2 ¹ © 2 C D · §C · §D J xy ¨ 1 1 cos 2T 2 1 cos 2T C 3 sin 2T ¸ ¨ 1 1 cos 2T 2 1 cos 2T D3 sin 2T ¸ 2 2 ¹ © 2 ¹ © 2 H xx ¨ (4) We know that H xcxc H ycyc J xcyc 2 H xx H yy 2 H xx H yy 2 H xx H yy 2 H xx H yy 2 H xx H yy 2 sin 2T cos 2T J xy sin 2T cos 2T J xy sin 2T J xy 2 (5) cos 2T Substituting eqn (5) in eqn (3) V xcxc B B § A1 · §A · 1 cos 2T 1 1 cos 2T C1 sin 2T ¸ H yy ¨ 1 1 cos 2T 1 1 cos 2T C1 sin 2T ¸ 2 2 © 2 ¹ © 2 ¹ B §A · J xy ¨ 1 sin 2T 1 sin 2T C1 cos 2T ¸ D1 2 © 2 ¹ (6) H xx ¨ Comparing eqn (4) and eqn (6) V xcxc V xcxc A B · §B · § A1 1 cos 2T 2 1 cos 2T A3 sin 2T ¸ H yy ¨ 1 1 cos 2T 2 1 cos 2T B3 sin 2T ¸ 2 2 ¹ © 2 ¹ © 2 C D · §D · §C J xy ¨ 1 1 cos 2T 2 1 cos 2T C 3 sin 2T ¸ ¨ 1 1 cos 2T 2 1 cos 2T D3 sin 2T ¸ 2 2 © 2 ¹ © 2 ¹ H xx ¨ (4) B B §A · § A1 · 1 cos 2T 1 1 cos 2T C1 sin 2T ¸ H yy ¨ 1 1 cos 2T 1 1 cos 2T C1 sin 2T ¸ 2 2 © 2 ¹ © 2 ¹ B §A · (6) J xy ¨ 1 sin 2T 1 sin 2T C1 cos 2T ¸ D1 2 © 2 ¹ H xx ¨ A1 B2 A B1 A2 B C1 A3 D1 D2 C3 A1 A2 2 D3 0 B3 C 2 D A B 2 C It should be noted that the corresponding transformation applied to V ycyc & V xcyc V xx AH xx BH yy CJ xy D V yy BH xx AH yy CJ xy D (7) A B V xy C H xx H yy J xy 2 Now let us consider the new coordinate system (x1, y1) which is related to the original coordinate system (x, y) by x1 x and y1 y Thus, the new coordinate system is a mirror reflection of the original system with respect to the y-axis. With reference to the new coordinate system, the velocity components are u1 u and v1 v Rates of strain and stresses are Hx x wu1 wx1 wu wx H xx Hy y wv1 wy1 wv wy H yy 1 1 1 1 Jx y 1 1 wv1 wu1 wx1 wy1 § wv wu · ¨¨ ¸¸ © wx wy ¹ Vx x V xx Vy y V yy Vx y V xy 1 1 1 1 1 1 Equations (8) and (9) into equation (7) (8) J xy (9) Vx x AH x1 x1 BH y1 y1 CJ x1 y1 D Vy y BH x1 x1 AH y1 y 1 CJ x1 y 1 D 1 1 1 1 Vx y 1 1 C H x1 x 1 H y1 y1 (10) A B J x1 y1 2 According to assumption (b), the relations between stress components and rates of strain components must be invariant to a coordinate transformation consisting of either a rotation or a mirror reflection of axes, Equation (7) and (10) independent of the coordinate system . Hence, C = 0 V xx AH xx BH yy CJ xy D V yy BH xx AH yy CJ xy D V xy A B J xy 2 C H xx H yy (7) According to assumption (c), Eqn. (7) gives D A B 2 p 2P The constant (A-B)/2 in the last equation of equation (10) is the proportionality constant connecting the shearing stress and rate of shearing strain which is generally denoted by the dynamic coefficient of viscosity, P The relations between stress and rate of strain in the two dimensional case given in equation (7) are reduced to V xx V yy V xy § wu wv · ¸¸ p 2 PH xx B¨¨ © w x wy ¹ § w u wv · ¸¸ p 2 PH yy B¨¨ © wx wy ¹ PJ xy 2PH xy B 2 P 3 The relations between stress and rate of strain can be extended to three dimensional flows. They are V xx V yy § wu wv ww · ¸¸ p 2 PH xx B¨¨ © wx wy wz ¹ § wu wv ww · ¸¸ p 2 PH yy B¨¨ © w x wy wz ¹ V xy § wu wv ww · ¸¸ p 2 PHzz B¨¨ © w x wy wz ¹ PJ xy 2PH xy V yx V yz PJ yz 2 PH yz V zy V zx PJ zx 2 PHzx V xz V zz The sum of the three normal stresses is V xx V yy V zz 3 p 2P 3 B .q For an incompressible fluid, .q 0 § ¨¨ .q © wu wv ww · ¸¸ wx wy wz ¹ The sum of the three normal stresses is V xx V yy V zz 3 p 2P 3 B .q For an incompressible fluid, .q V xx V yy V zz 3 0 p § ¨¨ .q © wu wv ww · ¸¸ wx wy wz ¹
