Jun. 05, 015 Chapter 6. Differential Analysis of Fluid Flow 6.1 Fluid Element Kinematics In this section, mathematical description of the motion of fluid elements moving in a flow field is given. A small fluid element in the shape of a cube will move to another position during Figure 6.1. t as in Figure 6.1 Type of motion and deformation for a fluid element 6.1.1 Velocity and acceleration fields revisited In rectangular coordinates, the fluid velocity is given by its components, i.e., V uivjw k (6.1) where u, v, and w are velocity components in x, y, and z directions, respectively. In terms of the velocity, the acceleration is given by V V V V a u v w t x y z (6.) or u u u u ax u v w t x y z v v v v ay u v w t x y z (6.3a) (6.3b) 1
w w w w az u v w t x y z (6.3c) The acceleration is concisely expressed as DV a (6.4) Dt where the operator D() () () () () u v w Dt t x y z (6.5) is termed the material derivative or substantial derivative. In vector notation, D() () V () Dt t (6.6) where i j k x y z (6.7) 6.1. Linear motion and deformation Consider a linear deformation of a fluid element. The original volume of this element is V xyz. The corresponding change in the original volume would be u V x yz t x Change in the rate at which the volume V is changing per unit volume due to the gradient u/ x is / 1 d V u x t u lim V dt t x Considering the other directions results in
1 d V u v w V V dt x y z The rate of the change of the volume per unit volume is the volumetric deformation rate. Figure 6.3 Linear deformation of a fluid element 6.1.3 Angular motion and deformation During t, the line segments OA and OB will rotate through the angles and to the new positions OA and OB. Then, the angular velocity is so that OA lim t0 t v/ x xt v tan t x x OA v/ x t v lim t0 t x If v/ x is positive, OA will be counterclockwise. Similarly, we have 3
OB u/ y t u lim t0 t y In this case, if u/ y is positive, OB The rotation about the z-axis will be clockwise. z is defined as the average of angular velocities of the two mutually perpendicular lines OA and OB. That is, OA and OB 1 v u z x y (6.1) Similarly, x y 1 w v y z 1 u w z x (6.13) (6.14) The three components can be combined to give the rotation vector such as i j k (6.15) x y z which is one half the curl of the velocity vector. That is, or 1 1 curlv V (6.16) 1 1w v 1u w 1v u V= i+ j k y z z x x y The vorticity is defined as a vector that is twice the rotation vector. That is, ξ V (6.17) The use of the vorticity to describe the rotational characteristics of the fluid simply eliminate the 1/ factor associated with the rotation vector. 4
Figure 6.4 Angular motion and deformation of a fluid element 6. Conservation of Mass The mass M of a system is constant as the system moves through the flow field. In equation, DM sys Dt 0 The control volume representation of the conservation of mass is given by dv Vn da 0 t (6.19) CV CS The first integral on the LHS of the above equation is the rate at which the mass within the control volume is changing and the second is the net rate at which the mass is flowing out through the control surface (rate of mass outflow rate of mass inflow). 6..1 Differential Form of Continuity Equation 5
Figure 6.5 A differential element for the development of conservation of mass equation Consider a control volume that is a small, stationary cubic element. The volume integral in Eq.(6.19) is given by t CV dv xyz t If we let u represent the x-component of the mass flow rate per unit area at the left face, then on the right face u u x x Therefore, the net rate of mass outflow in the x-direction is x u x y z Similarly, the net rates of mass outflows in the y- and z-directions are, respectively, y v z x y z w x y z Thus, the second integral of Eq.(6.19) is given by CS u v w Vn da x y z 6 xy z
Therefore, u v w 0 t x y z (6.7) In vector notation, V 0 t (6.8) For incompressible fluids, V 0 (6.30) or u v w 0 x y z (6.31) which applies to both steady and unsteady flow of incompressible fluids. 6..3 The Stream Function Steady, incompressible, plane, D flow represents one of the simplest types of flow of practical importance. For this flow, the continuity equation reduces to u v 0 x y (6.36) We can relate the two velocities using the stream function defined by u y ; v (6.37) x then the continuity equation is identically satisfied. That is, 0 x y y x 7
Thus, the use of the stream function means that the conservation of mass is satisfied. Another advantage of using the stream function is that lines along which is constant are streamlines. From the definition of streamline, dy v dx u The change of we move from one point to another is given by d dx dyvdxudy x y Along a line of constant, we have d 0 so that vdx udy 0 which is the equation for the streamline. The inflow dq crossing the arbitrary surface of AC in the figure below is dq udy vdx (6.38) or dq dy dx d y x (6.39) Thus, the volume rate of flow q between two streamlines is given by q (6.40) d 1 1 8
Figure 6.8 The flow between two streamlines In cylindrical polar coordinates, the continuity reduces to rv v 1 r 1 r r r 0 (6.41) with the stream function defined by v r 1 r ; v (6.4) r 6.3 Conservation of Linear Momentum To develop differential momentum equation, we can start the linear momentum equation such as F DP Dt (6.43) sys where F is the resultant force acting on a fluid mass and P is the linear momentum defined by P V dm sys In the previous chapter, it was shown that the sum of the force components on the control volume is given by F VdV VVndA t CV CS (6.44) Either Eq.(6.43) or Eq.(6.44) can be used to obtain the differential form of the linear momentum 9
equation. If we use the system approach, Eq.(6.43) by considering the differential mass V F D m Dt where F is the resultant force acting on m. Since m is constant, we have DV F m Dt m, then Thus, F ma (6.45) which is Newton s second law applied to m. 6.3.1 Description of Forces Acting on the Differential Element Two types of forces are considered here. They are body force and surface force. The body force is the weight of the element expressed as F b mg (6.46) Surface forces act on the element as a result of its interaction with its surroundings. Figure 6.10 Double script notation for stresses 10
Summing all forces in the x-direction yields F xx x y z xx yx zx xyz (6.48a) Similarly, F xy x y z xy yy zy xyz (6.48b) F xz zz x y z xz yz xyz (6.48c) The resultant surface force can now be expressed as F F i F j+ F k (6.49) s sx sy sz and the resultant force is FFs Fb where F b is the body force. Figure 6.11 Surface forces in the x direction acting on a fluid element 6.3. Equation of Motion Eq.(6.45) can be written as 11
F x F y ma x ma y F z ma z where m x y z. Using the expressions for the acceleration, u u u u x y z t x y z xx yx zx gx u v w v v v v x y z t x y z xy yy zy gy u v w w w w w x y z t x y z xz yz zz gz u v w (6.50a) (6.50b) (6.50c) which are the general differential equations of the motion for a fluid. 6.4 Inviscid Flow If we ignore the viscosity of fluids, then the shear stress can be negligible. For fluid in which there are no shear stresses, the normal stress at a point is independent of direction. If we define the pressure as the negative normal stress, i.e., p xx yy zz The negative sign denotes a compressive normal stress. 6.4.1 Euler s Equation of Motion For an inviscid flow, when replacing normal stresses by p, the equations of motion reduces to p u u u u gx u v w x t x y z 1
p v v v v gy u v w y t x y z p w w w w gz u v w z t x y z which are referred to as Euler s equation of motion. In vector notation, the Euler equation is expressed as V g p VV t (6.5) 6.4. The Bernoulli Equation For steady flows, the Euler Equation becomes gp V V (6.53) We want to integrate the differential equation. The acceleration of gravity vector is given by g= g z where g is the magnitude of gravity vector. Eq.(6.53) can be expressed as gzp VV-V V where the following vector identity is used: 1 VV VV-VV The equation of motion is now in the form of p 1 V g z V V We next take the dot product of each term with ds along a streamline. 13
p 1 ds V ds g z ds V V ds (6.54) Since ds and V are parallel and V V 0 VV ds is perpendicular to V. So, Thus, we have dp 1 d V gdz 0 (6.55) Eq.(6.55) can now be integrated to give dp V gz constant (6.56) For ideal fluids (inviscid and incompressible fluids), p V gz constant (6.57) It should be emphasized that the Bernoulli equation is restricted to (1) inviscid flow, () incompressible flow, (3) steady flow, and (4) flow along a streamline. 6.4.3 Irrotational Flow For rotation about z axis to be zero, it follows Therefore, 1 v u z x y = 0 v u x y (6.59) Similarly, 14
w v y z (6.60) u z w x (6.61) 6.4.4 The Bernoulli Equation for Irrotational Flow For irrotational flow, the RHS of Eq.(6.54) is zero regardless of the direction of ds. Integration of Eq.(6.55) again yields dp V gz constant (6.6) For ideal fluids (inviscid and incompressible fluids), p1 V1 p V z1 z (6.57) g g It should be emphasized that the Bernoulli equation is restricted to (1) inviscid flow, () incompressible flow, (3) steady flow, and (4) irrotational flow. 6.4.5 Velocity Potential For irrotational flows, the velocity components can be expressed in terms of a scalar function (, x yzt,,) as u ; v x ; w y (6.64) z where is called the velocity potential. In vector form, V (6.65) For an incompressible fluid, the conservation of mass is given by V=0 15
and for incompressible, irrotational flow, it follows that 0 (6.66) where is the Laplacian operator. Eq.(6.66) in cylindrical coordinate is 1 1 r 0 r r r r z (6.71) 16