MEE 309 Engineering Fluid Mechanics Lecture Notes PDF

## MEE 309: ENGINEERING FLUID MECHANICS ### Introduction Fluid mechanics is the discipline within applied mechanics that studies the behavior of liquids and gases at rest or in motion. > **Example:** A vast array of problems vary from the study of blood flow in capillaries to the flow of crude o...

## MEE 309: ENGINEERING FLUID MECHANICS ### Introduction Fluid mechanics is the discipline within applied mechanics that studies the behavior of liquids and gases at rest or in motion. > **Example:** A vast array of problems vary from the study of blood flow in capillaries to the flow of crude oil through a pipe. **Fluid mechanics helps explain:** * How a rocket generates thrust without air in outer space. * Why supersonic airplanes are inaudible until they pass you. * How a river flows downstream with significant velocity despite a shallow slope. * How data gathered from model airplanes informs the design of real things. * How aerodynamic car and truck design can improve gas mileage. **This makes fluid mechanics a very important and practical subject.** ### Fluid Kinematics Fluid kinematics is a branch of fluid mechanics that studies position, velocity, and acceleration of a fluid, describing and visualizing its motion without examining the forces that cause it. **In general, fluids flow, creating net motion of molecules from one point in space to another as a function of time.** * A typical portion of fluid contains so many molecules that tracking individual motion is unrealistic. * **Instead**, we consider **fluid particles** that interact with each other and surroundings. * This fluid particle motion can be described using terms of **velocity and acceleration**. ### Particle Location * **At a given instant**, a fluid property (density, pressure, velocity, acceleration) can be described as a function of its location. * This is called a **field representation** of the flow. * Different times result in different representations, so describing a flow requires a function of both spatial coordinates and time ($x$, $y$, $z$, $t$). ### Flow Descriptions There are two general approaches to analyzing fluid mechanics problems. #### Eulerian method * Uses the field concept. * Fluid motion is described by functions of space ($x$, $y$, $z$) and time ($t$) using properties like pressure, density, velocity. * Provides information about the flow at fixed points in space as fluid passes by. #### Lagrangian method * Involves following individual fluid particles as they move and determining changes in their properties over time. * Fluid particles are "tagged" and analyzed. > **Example:** Smoke discharging from a chimney. An Eulerian method fixes a measuring device at the top of the chimney to record temperature as a function of time, observing different particles passing by. A Lagrangian method attaches a device to a specific fluid particle to track its temperature as it moves. **Eulerian method is typically easier to use for experimental and analytical fluid mechanics investigations.** > **However,** there are applications where the Lagrangian approach is more convenient (for example, numerical fluid mechanics calculations involving interactions among particles). ### Velocity Field **One of the most important fluid variables is the velocity field:** $V = u(x, y, z, t)i + v(x, y, z, t)j + w(x, y, z, t)k$ * **u, v, and w are the x, y, and z components of the velocity vector.** * **Velocity is the time rate of change of the position vector for the fluid particle.** * **Velocity vector V has both direction and magnitude.** * **V = speed of the fluid** * **A velocity field can be visualized by sketching the velocity vector at different locations.** * **A stagnation point occurs where V = 0.** ### Acceleration Field Fluid motion can be described by either Lagrangian (following individual particles) or Eulerian (observing particles passing a fixed point) methods. * **Eulerian method requires expressing the acceleration field as a function of space and time.** * **The acceleration field is a resultant vector whose scalar components are:** * $a_x = \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w\frac{\partial u}{\partial z}$ * $a_y = \frac{\partial v}{\partial t} + u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} + w\frac{\partial v}{\partial z}$ * $a_z = \frac{\partial w}{\partial t} + u\frac{\partial w}{\partial x} + v\frac{\partial w}{\partial y} + w\frac{\partial w}{\partial z}$ * **Material derivative (substantial derivative) can be used to express the acceleration in shorthand form:** * $a = \frac{DV}{Dt} = \frac{\partial V}{\partial t} + (V \cdot \nabla)V$. ### Local and Convective Acceleration * **Local acceleration** refers to time-based changes in velocity at a fixed location. * $a_{local} = \frac{\partial V}{\partial t}$ * **Convective acceleration** results from changes in velocity due to movement of the fluid particle. * $a_{convective} = (V \cdot \nabla)V$ * **Total acceleration is the sum of local and convective acceleration.** * $a = a_{local} + a_{convective}$. ### Tangential and Normal Acceleration * **Tangential acceleration (a_s)** refers to the acceleration of a particle moving along a curved path. * **Normal acceleration (a_n)** is the acceleration towards the center of the curved path. * $a_n = \frac{V^2}{r}$ **This gives us an acceleration vector: $a = a_s + a_n$** ### One, Two, and Three-Dimensional Flows Fluid flow is generally three-dimensional and time-dependent. > **However,** certain assumptions can simplify analysis while maintaining accuracy. #### One-Dimensional Flow * Two velocity components are negligible. * Mathematically: $V = ui, v = 0, w = 0$ * **Example:** Flow in a pipe where average parameters are used. #### Two-Dimensional Flow * One velocity component is relatively small and negligible. * $V = ui + vj$ * **Example:** Flow between parallel plates, flow in the main stream of a wide river. #### Three-Dimensional Flow * Velocity is a function of time and three mutually perpendicular directions. * $V = V(x, y, z, t) = ui + vj + wk$ * **Example:** Flow in converging or diverging pipes. ### Steady and Unsteady Flow #### Steady Flow * Fluid characteristics (velocity, pressure, density) don't change with time at a given point in space -- they have a constant flow rate. * **Mathematically:** $\frac{\partial u}{\partial t} = 0, \frac{\partial v}{\partial t} = 0, \frac{\partial w}{\partial t} = 0$. * **Example:** Flow through a conduit with constant size and shape, and a velocity equation in the form $u = ax^2 + bx + c$ #### Unsteady Flow * Fluid characteristics change over time at a given point in space. * **Mathematically:** $\frac{\partial u}{\partial t} \neq 0, \frac{\partial v}{\partial t} \neq 0, \frac{\partial w}{\partial t} \neq 0$ **In reality, most flows are unsteady.** This is because it is generally harder to analyze and experiment with unsteady flows. So, steady-state assumptions are useful when they don't compromise the accuracy of the results. ### Compressible and Incompressible Flows * **Compressible flow:** Density of the fluid varies from point to point. * **Mathematically:** $\rho \neq constant$ * **Example:** Flow of gases through orifices, nozzles, gas turbines. * **Incompressible flow:** Density of the fluid remains constant throughout. * **Mathematically:** $\rho = constant$ * **Example:** Subsonic aerodynamics. ### Continuity Equation * The continuity equation is based on the principle of conservation of mass which states that, for a fluid flowing through a pipe, the mass passing through different sections across a length remains unchanged, unless additional fluid is added or removed. * **mathematically:** $\rho_1A_1V_1 = \rho_2A_2V_2$, where: * $A$ is the area of the pipe. * $V$ is the velocity of the fluid. * $\rho$ is the density of the fluid. * **For an incompressible fluid $\rho_1 = \rho_2$, simplifying the equation:** $A_1V_1 = A_2V_2$. ### Continuity Equation in Cartesian Coordinates * **Consider a fluid element in the shape of a parallelepiped:** with sides $dx$, $dy$, $dz$. * **Let:** * $\rho$ = Mass density of the fluid at an instant. * u, v, w = velocity components of the fluid across the parallelepiped's faces. * **Steps to derive the continuity equation in Cartesian coordinates:** 1. **Fluid influx:** * $\rho udydz$ 2. **Fluid efflux:** * $\rho udydz + \frac{\partial (\rho u)}{\partial x}dxdydz$ 3. **Gain in mass per unit time in X-direction:** * $- \frac{\partial (\rho u)}{\partial x}dxdydz$ 4. **Gain in mass per unit time in Y and Z directions:** * $-\frac{\partial (\rho v)}{\partial y} dxdydz$ * $-\frac{\partial (\rho w)}{\partial z} dxdydz$ 5. **Total gain in mass per unit time (along coordinate axes):** * $-\left(\frac{\partial (\rho u)}{\partial x}+\frac{\partial (\rho v)}{\partial y}+\frac{\partial (\rho w)}{\partial z}\right)dxdydz$ 6. **Rate of change in mass of the parallelepiped**: * $\frac{\partial (\rho dxdydz)}{\partial t}$ 7. **Equate steps 5 and 6**: * $-\left(\frac{\partial (\rho u)}{\partial x}+\frac{\partial (\rho v)}{\partial y}+\frac{\partial (\rho w)}{\partial z}\right)dxdydz = \frac{\partial (\rho dxdydz)}{\partial t}$ 8. **Simplify the equation:** * $\frac{\partial \rho}{\partial t} + \frac{\partial (\rho u)}{\partial x}+\frac{\partial (\rho v)}{\partial y}+\frac{\partial (\rho w)}{\partial z} = 0$ 9. **For steady flow:** $\frac{\partial \rho}{\partial t} = 0$ * $\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z} = 0$ 10. **For two-dimensional flow ($w = 0$):** * $\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y} = 0$ * **Integrating the above equation for a constant flow rate:** $Q = au = constant$ ### Types of Flow Lines Flow analysis can be simplified by visualizing and analyzing flow fields using: * **Streamlines:** A line everywhere tangent to the velocity field. * For steady flows, streamlines are fixed. * For unsteady flows, streamlines can change shape over time. * **Equation of streamline:** * $\frac{dx}{u} = \frac{dy}{v} = \frac{dz}{w}$ * **Properties of streamlines:** * Streamlines cannot intersect each other or themselves. * Fluid cannot move across streamlines. * Streamline spacing is inverse to flow velocity: flow converges as spacing decreases. * Streamline represents the direction of flow particles at a given instant. * In steady flow, streamlines and pathlines are identical. > **However**, in unsteady flow, streamline patterns may change over time. * **Stream tube:** A fluid mass bounded by a group of streamlines. * **Example:** Pipes and nozzles. * **Stream function (:psi):** A function of space and time that gives velocities perpendicular to the direction of differentiation. * For two-dimensional flow: $\psi = f(x, y, t)$. * **Equation for steady flow:** $\frac{\partial^2\psi}{\partial x\partial y} = 0$ * **Streakline:** Continuous path taken by fluid particles passing a given point over time. * **Example:** Smoke emitted from a chimney. * **Vortex:** Streamlines form concentric circles. * **Two types:** * **Free vortex:** Rotating fluid without external forces. * **Forced vortex:** Rotating fluid due to external forces. * **Circulation (Γ):** Line integral of the velocity component tangent to a closed curve in a flow field. > **Example:** The area around a closed curve of a vortex. * **Vorticity (ζ):** A vector that is twice the rotation vector. * **For two-dimensional flow in the xy plane:** $\zeta = 2\omega_{z}= \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}$. * **Rotational flow:** Flow possessing vorticity. * **Irrotational flow:** Rotation is equal to zero. * **Velocity potential (Φ):** A scalar function whose negative derivative in any direction gives the velocity in that direction. * **For steady flow:** $\Phi = f(x, y,z)$ * **Formula:** $u = -\frac{\partial \Phi}{\partial x}$, $v = -\frac{\partial \Phi}{\partial y}$, $w = -\frac{\partial \Phi}{\partial z}$. * **Laplace's equation (for incompressible, irrotational flow):** $\frac{\partial^2 \Phi}{\partial x^2} + \frac{\partial^2 \Phi}{\partial y^2} + + \frac{\partial^2 \Phi}{\partial z^2} = 0$ > **This type of flow is called potential flow.** * **Potential flow can be defined by boundary conditions**. * The boundary conditions are used to determine the velocity potential function and the velocity at points in the flow. **The velocity potential is a consequence of the irrotational nature of the flow, whereas the stream function is a consequence of conservation of mass.** **Stream function is defined for two-dimensional flow, while the velocity potential can be defined for three-dimensional flows.**