Differential Analysis of Fluid Flow Part I Governing

Differential Analysis of Fluid Flow Part I Governing Equations

Chapter review • Fluid element kinematics – Velocity and acceleration fields – Linear and angular motion and deformation • Continuity equation – Stream function • Equation of motion • Inviscid flow – – Euler equation of motion Bernoulli equation Irrotational flow. Velocity potential Some basic potential flow, plane potential flow • Viscous flow – Navier-Stokes equations. – Simple solutions for viscous incompressible fluids

Fluid Element Kinematics Types of motion and deformation for a fluid element

Fluid Element Kinematics Types of motion and deformation for a fluid element

Velocity and Acceleration Fields Velocity Acceleration

Linear Motion and Deformation Translation Simplest type of motion There are no velocity gradients y x

Linear Motion and Deformation Linear Deformation is caused by variations in the velocity in direction of velocity

Linear Motion and Deformation Linear Deformation is caused by variations in the velocity in direction of velocity Volumetric dilatation rate is a rate of change of volume per unit volume (details)

Linear Motion and Deformation Linear Deformation is caused by variations in the velocity in direction of velocity Volumetric dilatation rate is a rate of change of volume per unit volume (details) For incompressible fluid VDR equals zero Does it mean that if VDR is zero then fluid is incompressible?

Angular Motion and Deformation Rotation vector Vorticity

Angular Motion and Deformation Rotation vector Vorticity

Angular Motion and Deformation Analyze

Angular Motion and Deformation Analyze Fluid element will rotate about axis z as an undeformed block only when Otherwise the rotation will be associated with angular deformation When the rotation about z axis is zero If vorticity is zero flow field is termed as irrotational

Example: For a certain two dimensional flow field the velocity vector is given by equation Is this flow irrotational?

Rate of Shearing Strain Angular deformation results in a change in a shape of element The change in the original right angle formed by the line OA and OB is termed the shearing strain, is positive when original right angle is decreasing The rate of change of is called the rate of shearing strain or the rate of angular deformation

Conservation of Mass

Conservation of Mass Conservation of mass requires that the mass of a system remain constant For control volume We apply the last equation to the infinitesimal control volume to obtain the differential form of the continuity equation

Differential Form of Continuity Equation

Differential Form of Continuity Equation For infinitesimal control volume figure

Continuity Equation Differential form of continuity equation In vector notation Continuity equation is one of the fundamental equations of fluid mechanics and is valid for steady or unsteady flow, and compressible or incompressible fluids For steady flow of incompressible fluids

Example: The velocity components for a certain incompressible, steady flow field are Determine the form of the z component, w, required to satisfy the continuity equation

Example: The velocity components for a certain incompressible, steady flow field are Determine the form of the z component, w, required to satisfy the continuity equation Solution: Continuity equation for steady, incompressible flow For given velocity distribution so that Integration gives The third velocity component cannot be explicitly determined since the function f(x, y) can have any form and conservation of mass will still be satisfied.

Cylindrical Polar Coordinates

Cylindrical Polar Coordinates Velocity vector in cylindrical polar coordinates Continuity equation in cylindrical polar coordinates

Stream Function. Definition

Stream Function. Definition Introduction of a stream function is just a mathematical trick of replacing two variables (u and v) by a single higher-order function For steady, incompressible, plane (two-dimensional) flow the continuity equation reduces to Velocity components in plane flow field can be expressed in terms of a stream function, (x, y), which relates the velocities as In cylindrical coordinate continuity equation for incompressible, plane flow and velocity components

Stream Function. Properties • Whenever the velocity components are defined in terms of the stream function we know that conservation of mass will be satisfied. • Lines along which is constant are streamlines (details). If stream function is known one can plot family of streamlines to visualize the flow pattern. • The change in the value of the stream function is related to the volume rate of flow

Stream Function. Properties Consider two closely spaced streamlines

Stream Function. Properties Consider two closely spaced streamlines Volume flow rate per unit width Volume flow rate between two streamlines 1 and 2 If the 2 > 1, then q is positive, flow is from left to right If the 2 < 1, then q is negative, flow is from right to left

Example: The velocity components in a steady, incompressible, two-dimensional flow field are Determine the corresponding stream function and show on a sketch several streamlines. Indicate the direction of flow along the streamlines

Example: The velocity components in a steady, incompressible, two-dimensional flow field are Determine the corresponding stream function and show on a sketch several streamlines. Indicate the direction of flow along the streamlines Solution: From definition of stream function: Integration gives

Example: The velocity components in a steady, incompressible, two-dimensional flow field are Determine the corresponding stream function and show on a sketch several streamlines. Indicate the direction of flow along the streamlines Solution: Integration gives set C = 0, then for Streamlines are a family of hyperbolas with the = 0 streamlines as asymptotes

Example: In a certain steady, two-dimensional flow field the fluid density varies linearly with respect to the coordinate x; that is , ρ = Ax where A is a constant. If the x component of velocity u is given by equation u = y determine the expression for v

Example: In a certain steady, two-dimensional flow field the fluid density varies linearly with respect to the coordinate x; that is , ρ = Ax where A is a constant. If the x component of velocity u is given by equation u = y determine the expression for v Answer:

Conservation of Linear Momentum Resultant force acting on a fluid mass is equal to the time rate of change of linear momentum of the mass To develop the differential, linear momentum equations apply Newton’s second law of motion to the mass m

Forces Acting on the Differential Element Body forces are distributed through the element. We consider weight only. In component form

Forces Acting on the Differential Element Body forces are distributed through the element. We consider weight only. In component form

Forces Acting on the Differential Element Surface forces – result of interaction with surroundings. Surface forces acting on a fluid element can be described in terms of normal and shearing stresses. Normal stress Shearing stresses

Forces Acting on the Differential Element Surface forces – result of interaction with surroundings. Surface forces acting on a fluid element can be described in terms of normal and shearing stresses. Normal stress Shearing stresses

Stresses Notation and Sign Convention Stresses act on the planes parallel to coordinate plains First subscript – direction of normal to the plain on which stress acts Second subscript – direction of action of the stress Positive direction of the stress is defined as the positive coordinate direction on the surfaces for which the outward normal is in the positive coordinate direction. IF the outward normal points in the negative coordinate direction the stresses are considered positive if directed in the negative coordinate direction. (All stresses shown are positive)

Surface Forces

Surface Forces Surface forces acting on a small cubical fluid element Resultant surface force

Equations of Motion The resultant force acting on a fluid element must equal the mass times the acceleration of the element Equations of motion These are the general differential equations of motion for a fluid. They apply for any continuum (solid of fluid) in motion or at rest There are more unknowns than equations. Some additional information of stresses must be obtained

Inviscid Flow Shearing stresses develop in a moving fluid because of the viscosity of a fluid For inviscid or frictionless flow there are no shearing stresses and normal stress at a point is independent of direction, that is Pressure is a negative of the normal stress Negative sign is used so that a compressive normal stress gives positive value of pressure

Euler’s Equations of Motion For inviscid flow all shearing stresses are zero, normal stresses are replaced by –p Then general equation of motion reduce to: Euler’s equations of motion In vector form

Euler’s Equations of Motion For inviscid flow all shearing stresses are zero, normal stresses are replaced by –p Then general equation of motion reduce to: Euler’s equations of motion In vector form This is still unsolvable due to nonlinearity. But integrating these we can obtain Bernoulli equation

Bernoulli Equation For steady flow Select coordinate system with z axis vertical. Then Use vector identity Then Rearranging we have

Bernoulli Equation Take the dot product of each term with a differential length along a streamline Since has a direction along a streamline is perpendicular to. So And Then and are parallel. However the vector

Bernoulli Equation Thus equation (a) becomes (with and ) Integration gives For incompressible, inviscid, steady flow along a streamline (ideal fluid) In term of heads Bernoulli equation is restricted to inviscid, steady, incompressible flow along a streamline

End

Supplementary slides

Linear Motion and Deformation Linear Deformation The difference in velocity in x direction causes a “stretching” of volume element Rate of change of volume per unit volume due to gradient u/ x back

Differential Form of Continuity Equation back

Stream Function. Properties Lines along which is constant are streamlines Equation of a streamline: Change in the value of as we move from point (x, y) to point (x+dx, y+dy) Along line of constant we have Therefore, along a line of constant which is defining equation of a streamline back

Example: The stream function for an incompressible flow field is given by the equation where the stream function has the units of m 2/s with x and y in meters. (a) Sketch the streamlines passing through the origin. (b) Determine the rate of flow across the straight path AB shown in Figure