Tài liệu Chapter 4: Fluid Kinematics docx

Chapter 4: Fluid Kinematics Chapter 4: Fluid Kinematics ESOE 505221 Fluid Mechanics 2 Overview Fluid Kinematics deals with the motion of fluids without necessarily considering the forces and moments which create the motion. Chapter 4: Fluid Kinematics ESOE 505221 Fluid Mechanics 3 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). Chapter 4: Fluid Kinematics ESOE 505221 Fluid Mechanics 4 Lagrangian Description ( ) ( ) ( ) 0 0 0 0 0 0 0 0 0 , , , , , , , , , x x x y y y z z z du dx x x x y z t a u dt dt du dy y y x y z t u a dt dt dz du u a z z x y z t dt dt   =  = =          = ⇒ = ⇒ =          = =    =    Chapter 4: Fluid Kinematics ESOE 505221 Fluid Mechanics 5 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) Velocity field, Acceleration field, 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). ( ) ( ) ( ) , , , , , , , , ,V u x y z t i v x y z t j w x y z t k= + + r r r r ( ) ( ) ( ) , , , , , , , , , x y z a a x y z t i a x y z t j a x y z t k= + + r r r r ( ) , , ,a a x y z t= r r ( ) , , ,V V x y z t= r r Chapter 4: Fluid Kinematics ESOE 505221 Fluid Mechanics 6 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. Chapter 4: Fluid Kinematics ESOE 505221 Fluid Mechanics 7 Acceleration Field Consider a fluid particle and Newton's second law, The acceleration of the particle is the time derivative of the particle's velocity. However, particle velocity at a point at any instant in time t is the same as the fluid velocity, To take the time derivative of, chain rule must be used. particle particle particle F m a= r r particle particle dV a dt = r r ( ) ( ) ( ) ( ) , , particle particle particle particle V V x t y t z t= r r particle particle particle particle dx dy dz V dt V V V a t dt x dt y dt z dt ∂ ∂ ∂ ∂ = + + + ∂ ∂ ∂ ∂ r r r r r ,t) Chapter 4: Fluid Kinematics ESOE 505221 Fluid Mechanics 8 Acceleration Field Since 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. particle V V V V a u v w t x y z ∂ ∂ ∂ ∂ = + + + ∂ ∂ ∂ ∂ r r r r r , , particle particle particle dx dy dz u v w dt dt dt = = = Where ∂ is the partial derivative operator and d is the total derivative operator. Chapter 4: Fluid Kinematics ESOE 505221 Fluid Mechanics 9 EXAMPLE: Acceleration of a Fluid Particle through a Nozzle Nadeen is washing her car, using a nozzle. The nozzle is 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. How to apply this equation to the problem, particle V V V V a u v w t x y z ∂ ∂ ∂ ∂ = + + + ∂ ∂ ∂ ∂ r r r r r Chapter 4: Fluid Kinematics ESOE 505221 Fluid Mechanics 10 Flow Visualization Flow visualization is the visual examination of flow- field features. Important for both physical experiments and numerical (CFD) solutions. Numerous methods Streamlines and streamtubes Pathlines Streaklines Timelines Refractive techniques Surface flow techniques While quantitative study of fluid dynamics requires advanced mathematics, much can be learned from flow visualization [...]... 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... an individual particle ESOE 505221 Fluid Mechanics 20 Chapter 4: Fluid Kinematics Comparisons ESOE 505221 Fluid Mechanics 21 Chapter 4: Fluid Kinematics Timelines A Timeline is a set of adjacent fluid particles that were marked at the same (earlier) instant in time Timelines can be generated using a hydrogen bubble wire ESOE 505221 Fluid Mechanics 22 Chapter 4: Fluid Kinematics Timelines Timelines produced... 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: r r x = xstart + t ∫ r Vdt tstart ESOE 505221 Fluid Mechanics 15 Chapter 4: Fluid Kinematics Pathlines... the flow is called rotational ESOE 505221 Fluid Mechanics 31 Chapter 4: Fluid Kinematics Vorticity and Rotationality ESOE 505221 Fluid Mechanics 32 Chapter 4: Fluid Kinematics Comparison of Two Circular Flows Special case: consider two flows with circular streamlines ur = 0, uθ = ω r r 1  ∂ ( ruθ ) ∂ur  r 1  ∂ ( ζ=  − ÷ez =  r  ∂r ∂θ  r  ESOE 505221 Fluid Mechanics K r r ωr 2 r − 0 ÷ez = 2ωez... rate of shear strain ESOE 505221 Fluid Mechanics 25 Chapter 4: Fluid Kinematics Rate of Translation and Rotation To be useful, these rates must be expressed in terms of velocity and derivatives of velocity The rate of translation vector is described as the velocity vector In Cartesian coordinates: r r r r V = ui + vj + wk ESOE 505221 Fluid Mechanics 26 Chapter 4: Fluid Kinematics Rate of Translation... 505221 Fluid Mechanics 34 Chapter 4: Fluid Kinematics Reynolds—Transport Theorem (RTT) There is a direct analogy between the transformation from Lagrangian to Eulerian descriptions (for differential analysis using infinitesimally small fluid elements) and the transformation from systems to control volumes (for integral analysis using large, finite flow fields) ESOE 505221 Fluid Mechanics 35 Chapter 4: Fluid. .. 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. .. between x = - 2 m to 2 m and y = 0 m to 5 m; qualitatively describe the flow field ESOE 505221 Fluid Mechanics 24 Chapter 4: Fluid Kinematics Kinematic Description In fluid mechanics, an element may undergo four fundamental types of motion a) b) c) d) Translation Rotation Linear strain Shear strain Because fluids are in constant motion, motion and deformation is best described in terms of rates a) velocity:... ESOE 505221 Fluid Mechanics 12 Chapter 4: Fluid Kinematics Streamlines NASCAR surface pressure contours and streamlines ESOE 505221 Fluid Mechanics 13 Airplane surface pressure contours, volume streamlines, and surface streamlines 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... thumb for rotation ESOE 505221 Fluid Mechanics 27 Chapter 4: Fluid Kinematics Linear Strain Rate Linear Strain Rate is defined as the rate of increase in length per unit length In Cartesian coordinates ε xx = ∂u ∂v ∂w , ε yy = , ε zz = ∂x ∂y ∂z Volumetric strain rate in Cartesian coordinates 1 DV ∂u ∂v ∂w = ε xx + ε yy + ε zz = + + V Dt ∂x ∂y ∂z Since the volume of a fluid element is constant for an . Chapter 4: Fluid Kinematics Chapter 4: Fluid Kinematics ESOE 505221 Fluid Mechanics 2 Overview Fluid Kinematics deals with the motion of fluids. smoke in an airflow. Chapter 4: Fluid Kinematics ESOE 505221 Fluid Mechanics 18 Streaklines Chapter 4: Fluid Kinematics ESOE 505221 Fluid Mechanics 19 Streaklines