6.1 Fluid Element Kinematics( 運動學 )

6.1 Fluid Element Kinematics(運動學) CH. 6 : Differential Analysis of Fluid Flow Composition of Fluid Motion 1] Translation(병진, 竝進) 2] Linear Deformation : dilatation(팽창, 膨脹) , volume change, stretching 3] Rotation(회전, 回轉) 4] Angular Deformation : deformation(변형, 變形), shear deformation

6.1.2 Linear Motion and Deformation 1./ Translation of a Fluid Element (see p 312 Fig. 6.2) ; no velocity gradient ; 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.

2./ Linear Deformation of a Fluid Element Volumetric Dilatation (體積膨脹率)(see p312 Fig. 6.3) • rate at which the volume is changing per unit volume due to the gradient ;

◈ General case in which, are also present ◈ ; volumetric dilatation rate (체적팽창률) * meaning ; rate of change of the volume per unit volume * ; for an incompressible fluid since the element volume cannot change without a change in fluid density ◈ Variations in the velocity in the direction of the velocity such as ㅡㅡㅡㅡㅡㅡㅡㅡㅡ simply cause a linear deformation(선형 변형) of the element in the sense that the shape of the element does not change. ◈ Cross derivatives, such as, and , will cause the element to rotate and generally to undergo an angular deformation(각변형), which changes the shape of the element.

6.1.3 Angular Motion and Deformation 1./ Rotation Vector(Angular Velocity) , Vorticity(渦度) Vector 1] angular velocity of line OA and OB :

2] Rotation Vector ; rotation of the element about the z axis ; average of the angular velocities and Similarly (let clockwise; -, counterclockwise ; +) 3] vorticity(와도, 渦度) vector :

2./ Irrotational Flow(非回轉流動), Rotational Flow(回轉流動) 3./ Note! ◈ We observe from Eq. 6.12 that the fluid element will rotate about the z axis as an undeformed block(i.e., ) only when Otherwise the rotation will be associated with an angular deformation. ◈ We also note from Eq. 6.12 that when the rotation about the z axis is zero. 4./ Angular Deformation ; In addition to the rotation associated with the derivatives x and , it is observed from Fig. 6.4b that these

derivatives can cause the fluid element to undergo an angular deformation, which results in a change in shape ofthe element. 1] Definition[1] ◈ The deformation of the fluid particle in Fig. 6.4 is the rate of change of the angle that line segment OA makes with line segment OB.. ◈ If OA is rotating with an angular velocity different from that of OB, the particle is deforming. ◈ The deformation is represented by the rate-of-strain tensor. 2] Rate of Shearing Strain (or rate of angular deformation) ◈ The rate of angular deformation is related to a corresponding shearing stress which causes the fluid element to change in shape.

◈ From Eq. 6.18 we note that if , the rate of angular deformation is zero, and this condition corresponds to the case in which the element is simply rotating as an undeformed block. ◈ The instantaneous angular velocity of a fluid particle is the average of the instantaneous angular velocities of two mutually perpendicular lines on the fluid particle

Linear Translation • All points in the element have the same velocity (which is only true if there are o velocity gradients), then the element will simply translate from one position to another.

Linear Deformation 1/2 • The shape of the fluid element, described by the angles at its vertices, remains unchanged, since all right angles continue to be right angles. • 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

Angular Rotation 1/4 The angular velocity of line OA For small angles CCW CW “-” for CCW

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 Defining irrotation

Angular Rotation 4/4

Vorticity • Defining Vorticity ζ whichis a measurement of the rotation of a fluid elementas 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 ξ（Xi）η（Eta） 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 deformation in zx plane

Example 6.1 <Sol.>

6.2 Conservation of Mass • 6.2.1 Differential Form of Continuity Equation • 1./ General Form : Continuity Equation • 2./ Steady flow of compressible flow • 3./ Incompressible Flow • ; applicable to both steady and unsteady flow of incompressible fluids

Conservation of Mass 2/5 • The CV chosen is an infinitesimal cube with sides of length x,  y, and  z. Net rate of mass Outflow in x-direction

Conservation of Mass 3/5 Net rate of mass Outflow in x-direction Net rate of mass Outflow in y-direction Net rate of mass Outflow in z-direction

Conservation of Mass 4/5 Net rate of mass Outflow The differential equation for conservation of mass Continuity equation

Conservation of Mass 5/5 • Incompressible fluid • Steady flow

6.2.2 Cylindrical Polar Coordinates • 1./ General Form : Continuity Equation • 2./ Steady flow of compressible flow • 3./ Incompressible Flow • ; applicable to both steady and unsteady flow of incompressible fluids

6.2.3 The Stream Function 1./ Stream Function 1] Application Range ; Incompressible Plane Fluid Flows, not applicable to general 3-dimensional flow 2] Stream Function (Psi ; 流動函數) ; scalar function 3] Continuity Equation for Plane Flow Thus whenever the velocity components are defined in terms of the stream function we know that conservation of mass will be satisfied.

4] Mathematical Description of a Streamline ;

5] The difference between any two streamlines is equal to the flow rate per unit depth between the two streamlines. - If the upper streamline, , has a value greater than the lower streamline, , then q is positive, which indicates that the flow is from left to right. For the flow is from right to left.

6] Remark ! * Although we still don't know what is for a particular problem, but at least we have simplified the analysis by having to determine only one unknown function, , rather than the two functions, and * Another particular advantage of using the stream function is related to the fact that lines along which is constant are streamlines. * is constant along a streamline Note that flow never crosses streamlines, since by definition the velocity is tangent to the streamline.

2) Polar Coordinates - Continuity Equation - Velocity Components

6.3 Conservation of Linear Momentum 6.3.1 Description of Forces Acting on the Differential Element ◈ The intensity of the force per unit area at a point in a body can thus be characterized by a normal stress and two shearing stresses, if the orientation of the area is specified. ◈ double subscript notation for stresses (see p325 Fig. 6.10) - The first subscript indicates the direction of the normal to the plane on which the stress acts, and the second subscript indicates the direction of the stress. - Thus, normal stresses have repeated subscripts, whereas the subscripts for shearing stresses are always different.

6.3.2 Equations of Motion - General Differential Equations of Motion for a Fluid where

6.4 Inviscid Flow Definition of Inviscid Flow (非粘性유동) ; ◈ Flow fields in which the shearing stresses are assumed to be negligible are said to be inviscid, nonviscous, or frictionless. ◈ In this case ; ◈ We know that for some common fluids, such as air and water, the viscosity is small, and therefore it seems reasonable to assume that under some circumstances we may be able to simply neglect the effect of viscosity (and thus shearing stresses).

6.4.1 Euler's Equations of Motion Euler Equation = 비점성유동의 운동방정식

6.4.2 The Bernoulli Equation • Basic Assumptions • inviscid flow + steady flow + incompressible flow • + flow along a streamline • 2. Derivation (see p328-9, Fig. 6.12) • 3. General Form for Inviscid Flow • 4. Bernoulli Equation Figure 6.12 (p. 290)The notation for differential length along a streamline.

6.4.3 Irrotational Flow Criterion - Remarks * Since the weight acts through the element center of gravity, and the pressure acts in a direction normal to the element surface, neither of these forces can cause the element to rotate. * Boundary Layer Near the boundary the velocity changes rapidly from zero at the boundary (no-slip condition) to some relatively large value in a short distance from the boundary.

This rapid change in velocity gives rise to a large velocity gradient normal to the boundary and produces significant shearing stresses, even though the viscosity is small. Of course if we had a truly inviscid fluid, the fluid would simply "slide" past the boundary and the flow would be irrotational everywhere. But this is not the case for real fluids, so we will typically have a layer (usually very thin) near any fixed surface in a moving stream in which shearing stresses are not negligible. This layer is called the boundary layer. Figure 6.13 (p. 292)Uniform flow in the x direction.

6.1 Fluid Element Kinematics( 運動學 )