Ch05-01.ppt

* FUNDAMENTALS OF FLUID MECHANICS Chapter 5 The Differential Forms of Fundamental Laws * MAIN TOPICS Fluid Element Kinematics Conservation of Mass Conservation of Linear Momentum Inviscid Flow Conservation of Energy Equation * 5.1 Fluid Element Kinematics Motion of a Fluid Element Fluid Translation: The element moves from one point to another. Fluid Rotation: The element rotates about any or all of the x,y,z axes. Fluid Deformation: Angular Deformation:The element’s angles between the sides change. Linear Deformation:The element’s sides stretch or contract. * Linear Translation 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 1/2 A change in the x dimension requires a nonzero value of A ……………… y A ……………… z * Linear Deformation 2/2 The change in length of the sides may produce change in volume of the element. The change in The rate at which the ?V is changing per unit volume due to gradient ?u/ ?x If ?v/ ?y and ?w/ ?z are involved Volumetric dilatation rate （扩张） * Angular Rotation 1/4 The angular velocity of line OA For small angles CW Counterclockwise, “+” for CCW “-” for CW * Angular Rotation 2/4 The rotation of the element about the z-axis is defined as the average of the angular velocities ?OA and ?OB of the two mutually perpendicular lines OA and OB. In vector form * Angular Rotation 3/4 Defining vorticity （涡量） Defining irrotation * Angular Rotation 4/4 * Vorticity Defining Vorticity ζ which is a measurement of the rotation of a fluid element as it moves in the flow field: In cylindrical coordinates system * Angular Deformation 1/2 Angular deformation of a particle is given by the sum of the two angular deformation Rate of shearing strain or the rate of angular deformation * Angular Deformation 2/2 The rate of angular deformation in xy plane The rate of angular deformation in yz plane The rate of angular de