Advanced Fluid Mechanics

Name: Advanced Fluid Mechanics
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Master Advanced Fluid Dynamics: In-Depth Concepts, Differential Equations, Dimensional Analysis & Practical Applications

Created byProf. Samer

Last updated 9/2020

English

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What you'll learn

Understand how the differential equation of conservation of mass and the differential linear momentum equation are derived and applied
Calculate the stream function and pressure field, and plot streamlines for a known velocity field
Obtain analytical solutions of the equations of motion for simple flow fields
Understand dimensional analysis and similarity, principle of dimensional homogeneity Pi theorem, non-dimensionalization of basic equations
Understand concepts of inviscid, low Reynolds number, high Reynolds number, laminar and turbulent flow.
Identify and discuss the features of external flow
Calculate boundary layer parameters for flow past a flat plate
Calculate the lift and drag forces for various objects

Course content

6 sections • 86 lectures • 19h 3m total length

The Acceleration Field of a Fluid13:34
Explore the acceleration field of a fluid through eulerian and lagrangian descriptions, deriving the material derivative and its local and convective terms in terms of velocity components.
Example 14:01
Compute acceleration from the velocity field V = 3 t i + x j + t y^2 k by separating convective and local terms to obtain a_x, a_y, a_z.
Example 23:26
Analyze the one-dimensional velocity distribution u = V0(1 + 2x/a) and convective acceleration a_x = u du/dx, then compute entrance acceleration in g's for V0 = 10 ft/s and a = 6 inches, about 37 g.
The Differential Equation of Mass Conservation10:44
We explore the differential form of the mass conservation equation from the integral control-volume balance, for Cartesian coordinates and cylindrical coordinates, and discuss steady and incompressible versus compressible flows.
Example 31:37
Solve example three by finding the condition on the velocity field that yields an incompressible flow and satisfies the continuity equation for mass conservation in advanced fluid mechanics.
Example 41:56
Apply the continuity equation for incompressible flow to determine the form of the velocity components and their dependence on x, y, z, and time.
Linear and Shear Strain Rates20:03
Explore linear and shear strain rates in a two-dimensional incompressible flow. Derive epsilon_xx, epsilon_yy, and epsilon_xy from velocity gradients and examine volumetric dilation.
Navier-Stokes Equation23:20
Derive the Navier-Stokes equations for incompressible Newtonian flow by applying the momentum balance to a control volume and incorporating pressure and viscous stresses.
Example 520:30
This lecture verifies a velocity field against continuity and momentum equations for incompressible flow, derives the pressure field, and shows a hydrostatic-like distribution as a valid solution.
Example 65:37
Analyze a gravity-driven laminar flow between two stationary plates with no pressure gradient, deriving a parabolic velocity profile W(x) under the no-slip boundary conditions W(±H)=0.
The Stream Function11:20
Show how the stream function psi captures 2D incompressible flow, with u=∂ψ/∂y and v=−∂ψ/∂x; psi constant lines are streamlines and their differences yield the volumetric flow rate per unit width.
Example 713:16
derive the stream function psi for a steady incompressible flow, determine velocity components, sketch streamlines for multiple psi values, and identify hyperbola with asymptotes to map flow direction.
The Stream Function in Cylindrical Coordinates11:51
Derives the stream function in cylindrical coordinates for a line vortex, verifies continuity for incompressible flow, and shows circular streamlines with tangential velocity equal to K over R.
Vorticity and Irrotationality17:42
explain how fluid elements rotate and deform, derive vorticity as the curl of velocity, and distinguish irrotational flow from rotational flow with deformation, including solid-body rotation.
Frictionless Irrotational Flow20:34
Explore frictionless, incompressible, steady flow and derive Bernoulli's equation along streamlines under negligible viscosity. Irrotational flow yields a single constant, while rotational flow allows different constants for each streamline.
Velocity Potential11:07
Explore the velocity potential phi for irrotational flow, relate velocity as the gradient of phi, and show streamlines orthogonal to potential lines, with spacing affecting speed.
Example 816:02
Analyze two-dimensional flow around a 90-degree corner, derive the velocity field, and show the flow is rotational with nonzero vorticity. Use Bernoulli between same-elevation points to relate pressures.
Example 94:31
This lecture solves example nine for flow between two parallel plates, derives the stream function from the velocity profile, and shows the flow is rotational with no velocity potential.
Example 105:22
analyze a two-dimensional incompressible flow, verify continuity, derive velocity potential phi = 2x + 2y, and show the pressure gradient in x at x = 2 ft is zero.
Potential Flow: Uniform Flow9:51
The lecture introduces potential flow, focusing on uniform flow as the first type under incompressible, irrotational conditions, showing straight, parallel streamlines and the associated stream function and potential function.
Potential Flow: Source and Sink8:44
Describe a source or sink in potential flow by modeling radial flow from the origin with zero tangential velocity, yielding concentric streamlines and a defined volume flux through the origin.
Potential Flow: Vortex20:19
Explore two vortex types in potential flow: the zero-rotation (three-vortex) and the rotation vortex. Demonstrate how circulation, velocity potential, and streamlines relate to singularities and nonzero circulation.
Potential Flow: Doublet24:09
Explore the doublet in potential flow, formed by a source and a sink, derive its velocity potential and velocity field via the Laplace equation, and visualize streamlines with Matlab.
Example 113:48
Explores a draining tank that forms a vortex, deriving the surface shape from a three-vortex velocity potential and Bernoulli’s principle, linking surface elevation to the vortex circulation.
Example 123:20
Compute the circulation around a rectangular path in a two-dimensional flow using line integrals of velocity components, then show the result is zero under the given conditions.
Example 1310:39
Examine a tornado-like rotating flow with solid body rotation in inner and outer regions, deriving the velocity profile and pressure distribution, revealing the eye as the point of minimum pressure.
Superposition of Potential Flows: Source in a Uniform Stream—Half-Body20:30
Explore the superposition of a uniform stream and a source to form a half-body, analyze stagnation points and streamlines, and relate source strength to body width and velocity.
Example 1410:09
flow over a half-body hill: a 40 mph wind accelerates to 47.4 mph above the origin, with a 100 ft elevation; Bernoulli and mass conservation show P2 < P1.
Superposition of Potential Flows: Rankine Ovals23:02
Examine the superposition of a source, a sink, and a uniform flow to form a Rankine oval, locate stagnation points, and explore how parameters shape the curve with Matlab.
Superposition of Potential Flows: Flow Around a Cylinder35:36
Explore potential flow around a circular cylinder by superposing uniform flow and a doublet, deriving velocity and pressure, and noting drag, zero lift, and dalembert paradox.
Superposition of Potential Flows: Flow Around a Rotating Cylinder27:30
Explore potential flow around a rotating cylinder via superposition of a fixed cylinder and a three-vortex system, derive velocity and pressure distributions, locate stagnation points, and reveal the Magnus effect.
Example 155:14
Analyze stagnation point flow near a flat plate and how a source at the stagnation point creates a bump of height h, deriving h^2 = A/(2π).
Example 168:47
Apply a dimensionless length relation to determine the geometry, yielding a length of 13.1 feet. Use a bisection root-finding method on f(h/a)=0 to find a thickness of 3.3 feet.
Example 176:45
Water flows around a six-foot-diameter bridge pier at 12 ft/s. Flow separation invalidates ideal fluid theory and yields a front pressure and uniform rear pressure, producing drag per unit length.

Couette Flow between a Fixed and a Moving Plate6:25
Derives the linear velocity profile for Couette flow between a fixed lower plate and a moving upper plate in a two-dimensional, incompressible, steady flow with no pressure gradient.
Flow Due to Pressure Gradient between Two Fixed Plates7:15
Explain how a pressure gradient between two fixed plates drives parabolic flow, yielding a velocity profile that is zero at the walls and maximum at the center.
Example 115:13
demonstrates flow between parallel plates driven by a pressure gradient, derives wall shear stress, analyzes vorticity and stream function, and relates average velocity to maximum velocity for a parabolic profile.
Fully Developed Laminar Pipe Flow26:16
Explain fully developed laminar pipe flow with a parabolic velocity profile, boundary layer concepts, and the link between pressure drop, volumetric flow, and wall shear.
Example 24:18
Calculate the average velocity from volumetric flow rate and pipe area, then verify laminar flow and apply Hagen–Poiseuille formulas to relate pressure drop and power required to drive the flow.
Flow between Long Concentric Cylinders12:21
Derives the tangential velocity profile for steady, axisymmetric flow between a rotating inner cylinder and a fixed outer cylinder, showing zero radial velocity and solving for V_theta(r) with boundary conditions.

Introduction17:43
The lecture shows how fluid force on a cylinder depends on diameter, velocity, density, and viscosity, and how Reynolds number enables dimensionless analysis and scalable model testing.
The Pi Theorem17:54
the lecture applies the Buckingham pi theorem to a five-variable fluid-force problem, producing two dimensionless groups and showing F scales as V^2 times a function of Reynolds number.
Example 19:30
Learn to rewrite centrifugal pump power as a dimensionless relation using Buckingham pi theorem. Derive flow and power coefficients and a Reynolds-like parameter for rotating centrifuges.
Example 215:41
Analyze incompressible pipe flow with non-slip walls and surface roughness, linking wall shear stress to friction, velocity profile, Reynolds number, and viscosity via dimensionless groups.
Example 312:13
Examine wind-driven deflection of a billboard on an elastic column, perform scale-model wind-tunnel testing, and apply the mlt system to relate variables and predict model deflection.

Reynolds Number Regimes14:18
Explore Reynolds number regimes in pipe flow, identifying laminar, transitional, and turbulent flows and the critical Reynolds number around 2300; learn how hydraulic diameter influences calculations.
The Entrance Region10:07
Explore how a boundary layer forms at pipe inlets, creating an entrance region where velocity rises from zero at the wall to center line maximum, then develops with shear stress.
Laminar Flow in Pipes8:00
Derive laminar fully developed pipe flow with a parabolic velocity profile. Relate Vmax to Vavg and express pressure drop and friction factor, yielding F = 64/Re.
Example 15:31
Determine the pumping power for oil flow in a 0.28 m pipe by calculating the volumetric flow, Reynolds number for laminar flow, friction factor, and pressure drop.
Turbulent Velocity Profile10:19
In turbulent flow, velocity follows one-seventh power law, yielding a flatter core and steep near-wall gradient with higher shear; friction factor depends on Reynolds number and relative roughness via Colebrook.
The Moody Chart6:34
Explore moody chart to read friction factor f from Reynolds number; laminar friction is inversely proportional to Reynolds number, transitions are unreliable, and fully rough behavior depends on relative roughness.
Example 29:26
Determine the pipe diameter in a 150 m smooth pipe by equating head loss to friction factor with turbulent flow and Reynolds number iteration, yielding a diameter of 0.267 m.
Example 39:29
Compute Reynolds number and friction factor using the Halland explicit formula to determine pressure drop and pumping power for steady turbulent flow in a 2-inch stainless steel pipe.
Example 46:27
Determine velocity and flow rate in a 0.3 m diameter, 100 m pipe with 8 m head loss, using an empirical friction factor and turbulent flow with Reynolds number considerations.
Minor Losses40:49
Explore minor losses in pipe flows, including sudden contractions and expansions, elbows, tees, valves, and entrance losses, and learn how loss coefficients determine head loss.
Example 519:26
Apply Bernoulli to a copper piping system to compute pressure at point one, incorporating major and minor losses, velocities, and friction factors from elbows, globe valve, and faucet.
Example 610:42
Solve example six: compute the pipe diameter for 26 ft^3/s flow with major and minor losses, using Halland correlation and Reynolds number, converging to d ≈ 1.63 ft.
Piping Systems with Pumps and Turbines13:33
Pumps add energy to the fluid and turbines extract energy, using the modified Bernoulli equation with head losses and efficiencies of motor, pump, turbine, and generator.
Series and Parallel Pipes4:42
Analyze three parallel pipes where the area-velocity product is constant. The total flow equals the sum of individual flows while head loss is the same across all pipes.
Three-Reservoir Junction6:16
Solve a three-reservoir junction using mass conservation at junction J and energy equations between reservoirs A, B, and C, accounting for major head losses and iteration.
Example 722:43
Solve for the total flow rate and branch flows in two parallel pipes pumped from a lower reservoir to a higher one, using energy equations, friction factors, Reynolds numbers, and iterative solution.
Example 818:59
An energy-based analysis of water flow from a reservoir through two series pipes, applying major and minor losses and an iterative solution to cubic meters per second discharge of 0.107.
Example 921:11
Apply the energy equation and head losses to a three-reservoir network, solving for inlet flows Q1, Q2, Q3 by mass conservation and iterative head assumptions.
Pipe Flowrate Measurement12:45
Explore pipe flowrate measurement with venturi, orifice, and nozzle meters, noting gradual contractions to minimize losses and how discharge coefficients and Reynolds number affect actual versus ideal flow.
Example 1010:23
Fluid mechanics example computes methanol flow through a 4 cm pipe and 3 cm orifice with a mercury manometer 11 cm head, iterating Cd to obtain Q ≈ 3.04e-3 m^3/s.

Boundary Layer on a Flat Plate7:32
Examine boundary layer on a flat plate; define Delta(X) as thickness to 99 percent of free stream; compare viscous effects below with negligible above via momentum integral.
Momentum Integral Estimates: Boundary Layer Thickness24:44
Using the momentum integral method for laminar flow over a flat plate, derive boundary layer thickness delta(x) with a parabolic profile, and relate momentum thickness and drag to Re_x.
Momentum Integral Estimates: Drag Force and Drag Coefficient11:05
Analyze the displacement thickness delta star produced by the growing boundary layer over a flat plate, and relate velocity profiles to the apparent wall.
Momentum Integral Estimates: Displacement Thickness11:13
Identify the displacement thickness Δ* as the distance streamlines are displaced by the boundary layer on a flat plate, with Δ* = δ/3 and an imaginary wall pushing fluid away.
The Boundary Layer Equations24:08
Develop the boundary layer equations for a thin, two-dimensional incompressible flow under no separation. Obtain the simplified continuity and x-momentum equations, with pressure treated as constant across the boundary layer.
Blasius Solution: PDE to ODE25:29
Derive the boundary layer equations and convert the partial differential equations to an ordinary differential equation via a similarity transformation, yielding a third-order nonlinear ode with boundary conditions.
Blasius Solution: The Numerical Shooting Method13:11
Explore the Blasius solution using the numerical shooting method, converting to a first-order system, applying boundary conditions, and verifying the boundary-layer profile with Matlab code.
Blasius Solution: Displacement Thickness, Drag and Drag Coefficient7:04
Apply the Blasius boundary-layer solution to determine displacement thickness and boundary-layer thickness, compute wall shear stress and the drag coefficient, and derive drag force by integrating over the flat plate.
Turbulent Boundary Layer Flow28:52
Describe the turbulent boundary layer over a flat plate, from laminar-to-turbulent transition, using the 1/7 power-law velocity profile, and connect momentum thickness to drag coefficient and wall shear stress.
Example 111:05
This lecture models a hydrofoil as a flat plate in seawater, estimating boundary layer thickness, Reynolds number, laminar–turbulent transition, and roughness effects to compute drag.
Example 211:43
Analyze a 40 cm square duct where boundary layer displacement thickness accelerates core flow; apply continuity and Bernoulli to compute exit velocity and pressure drop.
Example 310:28
solves for downstream pitot position X for laminar boundary layer over a flat plate using a pitot tube and an oil manometer to relate pressure to velocity.
Boundary Layers with Pressure Gradients25:29
Explore boundary layers under pressure gradients, detailing viscous and pressure drag, favorable versus adverse gradients, and flow separation over a sphere. Compare laminar and turbulent profiles and drag coefficient trends.
Drag Coefficient Data for Various Objects17:03
Explore how drag coefficients vary with shape, Reynolds number, and boundary layer behavior, from creeping laminar flow to turbulent wakes, and learn streamlining to reduce drag.
Flow over an Airfoil: Generation of Lift31:52
Explore how lift forms on an airfoil from pressure differences at angle of attack, with chord, span, thickness, camber line, and aspect ratio shaping lift, drag, and stall.
Example 49:57
Analyzes a submerged cube with specific gravity 1.8 in water, balancing weight, buoyancy, and drag to obtain terminal velocity for two orientations (about 2.04 m/s and 2.11 m/s).
Example 53:13
Advanced fluid mechanics: analyze how drag coefficient changes with windows and roof configurations, from 0.35 to about 0.425–0.45, and determine equal power speed, about 59.78 mph.
Example 69:54
Advanced fluid mechanics compares a square flat plate in cases a and b: case a has thicker boundary layer and higher drag than case b, whose boundary layer is thinner.
Example 78:16
Determine the outlet velocity that balances weight and drag on a 2.6 g, 3.8 cm ping pong ball using the Cd–Re curve, yielding 8.57 m/s and Reynolds number near 22,411.
Example 89:54
Compute the takeoff distance for a 75,000 lb aircraft using thrust, drag as a function of velocity, and lift coefficient; derive stall and takeoff speeds, yielding about 2441.44 ft.

Requirements

Physics and Calculus
Fundamentals of Fluid Mechanics Course

Description

Dive into our Advanced Fluid Mechanics course, designed as a continuation of our Fundamentals of Fluid Mechanics course. This comprehensive program covers essential topics and concepts, offering deeper insights and advanced applications in fluid dynamics.

In this course, you will explore:

Differential Relations: Gain a deeper understanding of fluid particles, fluid acceleration, the Continuity equation, Potential flows, and the Navier-Stokes equation.
Dimensional Analysis & Similarity: Learn about the principle of dimensional homogeneity, the Pi theorem, non-dimensionalization of basic equations, and the challenges of modeling.
Flow in Ducts & Boundary Layer Flows: Examine pressure drop calculations, minor losses in fittings, and the energy equation applied to pumps and turbines.
Flow Over Immersed Bodies: Delve into drag and lift calculations, essential for analyzing fluid motion and optimizing designs.
MATLAB Codes for Potential Flows: Access MATLAB codes, enhancing your computational skills in fluid dynamics and facilitating potential flow analysis.
Advanced Applications & Real-World Examples: Discover practical applications and real-world examples, enabling you to apply your knowledge to complex fluid dynamics problems.
Hands-On Learning & Problem Solving: Engage in hands-on learning and problem-solving activities to reinforce your understanding and develop your skills.
Expert Guidance & Support: Benefit from expert guidance and support throughout the course, ensuring a thorough understanding of the material and the ability to apply it effectively.

Enroll in our Advanced Fluid Mechanics course today to expand your knowledge and skills in fluid dynamics. Master the continuation of the Fundamentals of Fluid Mechanics and boost your expertise, opening up new opportunities in your academic and professional pursuits.

Who this course is for:

Engineering Students

Advanced Fluid Mechanics

What you'll learn

Explore related topics

Course content

Differential Relations for Fluid Flow34 lectures • 7hr 15min

Approximate Solutions of the Navier-Stokes Equation6 lectures • 1hr 12min

Dimensional Analysis and Similarity5 lectures • 1hr 13min

Viscous Flow in Ducts20 lectures • 4hr 22min

Flow Past Immersed Bodies20 lectures • 5hr 2min

Bonus Section: Coupons for all courses1 lecture • 1min

Requirements

Description

Who this course is for: