Chapter 4: Fluid Kinematics

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Chapter 4: Fluid Kinematics

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Fluid Kinematics deals with the motion of fluids without necessarily considering the forces and moments which create the motion. Two ways to describe motion are Lagrangian and Eulerian description. Lagrangian description of fluid flow tracks the position and velocity of individual particles. (eg. Brilliard ball on a pooltable.)

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Chapter 4: Fluid Kinematics
Overview Fluid Kinematics deals with the motion of fluids without necessarily considering the forces and moments which create the motion. ESOE 505221 Fluid Mechanics 2 Chapter 4: Fluid Kinematics
Lagrangian Description Two ways to describe motion are Lagrangian and Eulerian description Lagrangian description of fluid flow tracks the position and velocity of individual particles. (eg. Brilliard ball on a pooltable.) Motion is described based upon Newton's laws. Difficult to use for practical flow analysis. Fluids are composed of billions of molecules. Interaction between molecules hard to describe/model. However, useful for specialized applications Sprays, particles, bubble dynamics, rarefied gases. Coupled Eulerian-Lagrangian methods. Named after Italian mathematician Joseph Louis Lagrange (1736-1813). ESOE 505221 Fluid Mechanics 3 Chapter 4: Fluid Kinematics
Lagrangian Description  x = x ( x0 , y0 , z0 , t )  dx  du x  u x = dt ax = dt      dy  du y  y = y ( x0 , y0 , z0 , t ) ⇒ u y = ⇒ a y =   dt  dt   dz  du z  z = z ( x0 , y0 , z0 , t )  u z = dt az = dt   ESOE 505221 Fluid Mechanics 4 Chapter 4: Fluid Kinematics
Eulerian Description Eulerian description of fluid flow: a flow domain or control volume is defined by which fluid flows in and out. We define field variables which are functions of space and time. Pressure field, P=P(x,y,z,t) r r Velocity field, V = V ( x, y , z , t ) r r r r V = u ( x, y , z , t ) i + v ( x , y , z , t ) j + w ( x , y , z , t ) k r r Acceleration field, a = a ( x, y , z , t ) r r r r a = a x ( x , y , z , t ) i + a y ( x , y , z , t ) j + a z ( x, y , z , t ) k These (and other) field variables define the flow field. Well suited for formulation of initial boundary-value problems (PDE's). Named after Swiss mathematician Leonhard Euler (1707-1783). ESOE 505221 Fluid Mechanics 5 Chapter 4: Fluid Kinematics
Example: Coupled Eulerian-Lagrangian Method Forensic analysis of Columbia accident: simulation of shuttle debris trajectory using Eulerian CFD for flow field and Lagrangian method for the debris. ESOE 505221 Fluid Mechanics 6 Chapter 4: Fluid Kinematics
Acceleration Field Consider a fluid particle and Newton's second law, r r Fparticle = m particle a particle The acceleration of the particle is the time derivative of r the particle's velocity. r dV particle a particle = dt However, particle velocity at a point at any instant in time t is the same as the fluid velocity, r r V particle = V ( x particle ( t ) , y particle ( t ) , z particle ( t ) ) ,t) To take the time derivative of, chain rule must be used. r r r r r ∂V dt ∂V dx particle ∂V dy particle ∂V dz particle a particle = + + + ∂t dt ∂x dt ∂y dt ∂z dt ESOE 505221 Fluid Mechanics 7 Chapter 4: Fluid Kinematics
Acceleration Field Where ∂ is the partial derivative operator and d is the total derivative operator. Since dx particle dy particle dz particle = u, = v, =w dt dt dt r r r r r ∂V ∂V ∂V ∂V a particle = +u +v +w ∂t ∂x ∂y ∂z First term is called the local acceleration and is nonzero only for unsteady flows. Second term is called the advective acceleration and accounts for the effect of the fluid particle moving to a new location in the flow, where the velocity is different. ESOE 505221 Fluid Mechanics 8 Chapter 4: Fluid Kinematics
EXAMPLE: Acceleration of a Fluid Particle through a Nozzle How to applyrthis equation to the problem r r r Nadeen is washing her car, r ∂V ∂V ∂V ∂V using a nozzle. The nozzle is a particle = +u +v +w ∂t ∂x ∂y ∂z 3.90 in (0.325 ft) long, with an inlet diameter of 0.420 in (0.0350 ft) and an outlet diameter of 0.182 in. The volume flow rate through the garden hose (and through the nozzle) is 0.841 gal/min (0.00187 ft3/s), and the flow is steady. Estimate the magnitude of the acceleration of a fluid particle moving down the centerline of the nozzle. ESOE 505221 Fluid Mechanics 9 Chapter 4: Fluid Kinematics
Flow Visualization Flow visualization is the While quantitative study of fluid visual examination of dynamics requires advanced flow-field features. mathematics, much can be Important for both learned from flow visualization physical experiments and numerical (CFD) solutions. Numerous methods Streamlines and streamtubes Pathlines Streaklines Timelines Refractive techniques Surface flow techniques ESOE 505221 Fluid Mechanics 10 Chapter 4: Fluid Kinematics
Streamlines A Streamline is a curve that is everywhere tangent to the instantaneous local velocity vector. Consider an arc length r r r r dr = dxi + dyj + dzk r dr must be parallel to the local velocity vector r r r r V = ui + vj + wk Geometric arguments results in the equation for a streamline dr dx dy dz = = = V u v w ESOE 505221 Fluid Mechanics 11 Chapter 4: Fluid Kinematics
EXAMPLE C: Streamlines in the xy Plane—An Analytical Solution For the same velocity field of Example A, plot several streamlines in the right half of the flow (x > 0) and compare to the velocity vectors. where C is a constant of integration that can be set to various values in order to plot the streamlines. ESOE 505221 Fluid Mechanics 12 Chapter 4: Fluid Kinematics
Streamlines Airplane surface pressure contours, NASCAR surface pressure contours volume streamlines, and surface and streamlines streamlines ESOE 505221 Fluid Mechanics 13 Chapter 4: Fluid Kinematics
Streamtube A streamtube consists of a bundle of streamlines (Both are instantaneous quantities).  Fluid within a streamtube must remain there and cannot cross the boundary of the streamtube. In an unsteady flow, the streamline pattern may change significantly with time.⇒ the mass flow rate passing through any cross-sectional slice of a given streamtube must remain the same. ESOE 505221 Fluid Mechanics 14 Chapter 4: Fluid Kinematics
Pathlines A Pathline is the actual path traveled by an individual fluid particle over some time period. Same as the fluid particle's material position vector (x particle ( t ) , y particle ( t ) , z particle ( t ) ) Particle location at time t: t r r r x = xstart + ∫ Vdt tstart ESOE 505221 Fluid Mechanics 15 Chapter 4: Fluid Kinematics
Pathlines A modern experimental technique called particle image velocimetry (PIV) utilizes (tracer) particle pathlines to measure the velocity field over an entire plane in a flow (Adrian, 1991). ESOE 505221 Fluid Mechanics 16 Chapter 4: Fluid Kinematics
Streaklines A Streakline is the locus of fluid particles that have passed sequentially through a prescribed point in the flow. Easy to generate in experiments: dye in a water flow, or smoke in an airflow. ESOE 505221 Fluid Mechanics 17 Chapter 4: Fluid Kinematics
Streaklines ESOE 505221 Fluid Mechanics 18 Chapter 4: Fluid Kinematics
Streaklines Karman Vortex street Cylinder x/D A smoke wire with mineral oil was heated to generate a rake of Streaklines ESOE 505221 Fluid Mechanics 19 Chapter 4: Fluid Kinematics
Comparisons For steady flow, streamlines, pathlines, and streaklines are identical. For unsteady flow, they can be very different. Streamlines are an instantaneous picture of the flow field Pathlines and Streaklines are flow patterns that have a time history associated with them. Streakline: instantaneous snapshot of a time- integrated flow pattern. Pathline: time-exposed flow path of an individual particle. ESOE 505221 Fluid Mechanics 20 Chapter 4: Fluid Kinematics

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