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2. Fluid-Flow Equations Governing Equations

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2. Fluid-Flow Equations Governing Equations • Conservation equations for: ‒ mass ‒ momentum ‒ energy ‒ (other constituents) • Alternative forms: ‒ integral (control-volume) equations ‒ differential equations Integral (Control-Volume) Approach Consider the budget of any physical quantity in any control volume V V TIME DERIVATIVE ADVECTIVE + DIFFUSIVE FLUX SOURCE + = of amount in 푉 through boundary of 푉 in 푉 → Finite-volume method for CFD Mass Conservation (Continuity) Mass conservation: mass is neither created nor destroyed d u (mass) = net inward mass flux d푡 V un A d (mass) + net outward mass flux = 0 d푡 d mass + ෍ mass flux = 0 d푡 faces Mass in a cell: ρ푉 Mass flux through a face: 퐶 = ρu • A Mass Conservation - Differential Equation t Conservation statement: d z mass + net outward mass flux = 0 w n d푡 y b e d s ρ푉 + (ρ푢퐴) − (ρ푢퐴) + (ρ푣퐴) − (ρ푣퐴) + (ρ푤퐴) − (ρ푤퐴) = 0 x d푡 푒 푤 푛 푠 푡 푏 d ρΔ푥Δ푦Δ푧 + [(ρ푢) − ρ푢) Δ푦Δ푧 + [(ρ푣) − ρ푣) Δ푧Δ푥 + [(ρ푤) − ρ푤) Δ푥Δ푦 = 0 d푡 푒 푤 푛 푠 푡 푏 Divide by volume: dρ (ρ푢) − (ρ푢) (ρ푣) − (ρ푣) (ρ푤) − (ρ푤) + 푒 푤 + 푛 푠 + 푡 푏 = 0 d푡 Δ푥 Δ푦 Δ푧 dρ Δ(ρ푢) Δ(ρ푣) Δ(ρ푤) + + + = 0 d푡 Δ푥 Δ푦 Δ푧 Shrink to a point: 휕ρ 휕(ρ푢) 휕(ρ푣) 휕(ρ푤) 휕ρ + + + = 0 + ∇ • (ρu) = 0 휕푡 휕푥 휕푦 휕푧 휕푡 Continuity in Incompressible Flow t z w n b y s e d x (volume) + net outward volume flux = 0 d푡 휕푢 휕푣 휕푤 + + = 0 ∇ • u = 0 휕푥 휕푦 휕푧 Momentum Equation Momentum Principle: force = rate of change of momentum F If steady: force = (momentum flux)out – (momentum flux)in If unsteady: force = d/d푡(momentum inside control volume) + (momentum flux)out – (momentum flux)in Momentum Equation Rate of change of momentum = force d (momentum) + net outward momentum flux = force d푡 u V un A d mass × u + ෍ (mass flux × u) = F d푡 faces Momentum of fluid in a cell = mass × u = (ρ푉)u Momentum flux through a face = mass flux × u = (ρu • A)u Fluid Forces force Surface forces (proportional to area): stress = area • pressure y 휕푢 • viscous force: τ = μ 휕푦 U • reactions from boundaries force Body forces (proportional to volume): force density = z volume • gravity: −ρe푧 g axis 2 R R 2 • centrifugal force: ρΩ R r u −2ρΩ ∧ u • Coriolis force: In inertial frame In rotating frame Differential Equation Conservation statement: t d z (momentum) + net momentum flux = force d푡 w n y b e d s ρ푉푢 + (ρ푢퐴) 푢 − (ρ푢퐴) 푢 + (ρ푣퐴) 푢 − (ρ푣퐴) 푢 + (ρ푤퐴) 푢 − (ρ푤퐴) 푢 d푡 푒 푒 푤 푤 푛 푛 푠 푠 푡 푡 푏 푏 x = 푝푤퐴푤 − 푝푒퐴푒 + viscous and other forces d ρΔ푥Δ푦Δ푧 푢 + [(ρ푢) 푢 − ρ푢) 푢 Δ푦Δ푧 + [(ρ푣) 푢 − ρ푣) 푢 Δ푧Δ푥 + [(ρ푤) 푢 − ρ푤) 푢 Δ푥Δ푦 d푡 푒 푒 푤 푤 푛 푛 푠 푠 푡 푡 푏 푏 = 푝푤 − 푝푒 Δ푦Δ푧 + viscous and other forces Divide by volume: d(ρ푢) (ρ푢푢)푒 − (ρ푢푢)푤 (ρ푣푢)푛 − (ρ푣푢)푠 (ρ푤푢)푡 − (ρ푤푢)푏 푝푒 − 푝푤 viscous and + + + = − + d푡 Δ푥 Δ푦 Δ푧 Δ푥 other forces d(ρ푢) Δ(ρ푢푢) Δ(ρ푣푢) Δ(ρ푤푢) Δ푝 viscous and + + + = − + d푡 Δ푥 Δ푦 Δ푧 Δ푥 other forces Shrink to a point: 휕(ρ푢) 휕(ρ푢푢) 휕(ρ푣푢) 휕(ρ푤푢) 휕푝 + + + = − + μ∇2푢 + other forces 휕푡 휕푥 휕푦 휕푧 휕푥 General Scalar Time derivative + net outward flux = source ϕ = concentration (amount per unit mass) ρ푉ϕ Amount in a cell: (mass concentration) u Flux through a face: V un ‒ advection: (ρu • A)ϕ (mass flux concentration) A 휕ϕ ‒ diffusion: −Γ 퐴 휕푛 (diffusivity gradient area) Source: 푆 = 푠푉 (source density volume) d 휕ϕ mass × ϕ + ෍ (mass flux × ϕ − Γ 퐴) = 푠 푉 d푡 휕푛 faces 휕(ρϕ) 휕 휕ϕ 휕 휕ϕ 휕 휕ϕ + ρ푢ϕ − Γ + ρ푣ϕ − Γ + ρ푤ϕ − Γ = 푠 휕푡 휕푥 휕푥 휕푦 휕푦 휕푧 휕푧 Momentum Components as General Scalars Momentum equation: d 휕푢 mass × 푢 + ෍ mass flux × 푢 = ෍ ( μ 퐴) + other forces d푡 휕푛 faces faces viscous forces d 휕푢 mass × 푢 + ෍ mass flux × 푢 − μ 퐴 = other forces d푡 휕푛 faces General scalar-transport equation: d 휕ϕ mass × ϕ + ෍ mass flux × ϕ − Γ 퐴 = 푆 d푡 휕푛 faces • Velocity components 푢, 푣, 푤 satisfy individual scalar-transport equations: ‒ concentration, ϕ velocity component, 푢, 푣, 푤 ‒ diffusivity, viscosity, μ ‒ source, 푆 non-viscous forces • Differences: ‒ momentum equations are non-linear ‒ momentum equations are coupled ‒ the velocity field also has to be mass-consistent Differential Equations For Fluid Flow Forms of the equations in primitive variables may be: • Conservative ‒ can be integrated directly to give “net flux = source” • Non-conservative ‒ can’t be integrated directly Other forms of the equations include those for: • Derived variables ‒ e.g. velocity potential Example d (푦2) = (푥) conservative d푥 d푦 2푦 = (푥) non-conservative d푥 Same equation! ... but only the first can be integrated directly Rate of Change Following the Flow ϕ ≡ ϕ(푡, x) Total derivative dϕ 휕ϕ 휕ϕ d푥 휕ϕ d푦 휕ϕ d푧 ≡ + + + (following any path x(푡) (x(t), y(t), z(t)) d푡 휕푡 휕푥 d푡 휕푦 d푡 휕푧 d푡 Material derivative d푥 Dϕ 휕ϕ 휕ϕ 휕ϕ 휕ϕ = u ≡ + 푢 + 푣 + 푤 (following the flow): d푡 D푡 휕푡 휕푥 휕푦 휕푧 D 휕 휕 휕 휕 ≡ + 푢 + 푣 + 푤 D푡 휕푡 휕푥 휕푦 휕푧 D 휕 ≡ + u • ∇ D푡 휕푡 Non-Conservative Flow Equations conservative form non-conservative form 휕 휕 휕 휕 Dϕ (ρϕ) + (ρ푢ϕ) + (ρ푣ϕ) + (ρ푤ϕ) → ρ 휕푡 휕푥 휕푦 휕푧 D푡 (mass conservation) D푢 휕푝 ρ = − + μ∇2푢 e.g. momentum equation: D푡 휕푥 mass × acceleration forces 휕(ρϕ) 휕(ρ푢ϕ) 휕(ρ푣ϕ) 휕(ρ푤ϕ) Proof: + + + 휕푡 휕푥 휕푦 휕푧 휕ρ 휕ϕ 휕(ρ푢) 휕ϕ 휕(ρ푣) 휕ϕ 휕(ρ푤) 휕ϕ = ϕ + ρ + ϕ + ρ푢 + ϕ + ρ푣 + ϕ + ρ푤 휕푡 휕푡 휕푥 휕푥 휕푦 휕푦 휕푧 휕푧 휕ρ 휕(ρ푢) 휕(ρ푣) 휕(ρ푤) 휕ϕ 휕ϕ 휕ϕ 휕ϕ = + + + ϕ + ρ + 푢 + 푣 + 푤 휕푡 휕푥 휕푦 휕푧 휕푡 휕푥 휕푦 휕푧 =0 by continuity =Dϕ/D푡 by definition Dϕ = ρ D푡 Example Q1 (Equation Manipulation) In 2-d flow, the continuity and x-momentum equations can be written in conservative form as 휕ρ 휕 휕 휕 휕 휕 휕푝 + (ρ푢) + (ρ푣) = 0 (ρ푢) + (ρ푢푢) + (ρ푣푢) = − + μ∇2푢 휕푡 휕푥 휕푦 휕푡 휕푥 휕푦 휕푥 (a) Show that these can be written in the equivalent non-conservative forms: Dρ 휕푢 휕푣 D푢 휕푝 + ρ( + ) = 0 ρ = − + μ∇2푢 D푡 휕푥 휕푦 D푡 휕푥 (b) Define carefully what is meant by the statement that a flow is incompressible. To what does the continuity equation reduce in incompressible flow? (c) Write down conservative forms of the 3-d equations for mass and x-momentum. (d) Write down the 푧-momentum equation, including the gravitational force. (e) Show that, for constant-density flows, pressure and gravity can be combined in the momentum equations via the piezometric pressure 푝 + ρ푧. axis (f) In a rotating reference frame there are additional apparent forces (per unit volume): 2 R R 2 centrifugal force: ρΩ R r Coriolis force: −2ρΩ ∧ u where Ω is the angular velocity of the reference frame, u is the fluid velocity in that frame, r is the position vector and R is its projection perpendicular to the axis of rotation. By writing the centrifugal force as the gradient of some quantity show that it can be subsumed into a modified pressure. Also, find the components of the Coriolis force if rotation is about the 푧 axis. In 2-d flow, the continuity and x-momentum equations can be written in conservative form as 휕ρ 휕 휕 휕 휕 휕 휕푝 + (ρ푢) + (ρ푣) = 0 (ρ푢) + (ρ푢푢) + (ρ푣푢) = − + μ∇2푢 휕푡 휕푥 휕푦 휕푡 휕푥 휕푦 휕푥 (a) Show that these can be written in the equivalent non-conservative forms: Dρ 휕푢 휕푣 D푢 휕푝 + ρ( + ) = 0 ρ = − + μ∇2푢 D푡 휕푥 휕푦 D푡 휕푥 Continuity: Momentum: 휕ρ 휕 휕 휕 휕 휕 휕푝 + (ρ푢) + (ρ푣) = 0 (ρ푢) + (ρ푢푢) + (ρ푣푢) = − + μ∇2푢 휕푡 휕푥 휕푦 휕푡 휕푥 휕푦 휕푥 휕ρ 휕푢 휕ρ 휕ρ 휕푢 휕ρ 휕푣 LHS = 푢 + ρ + 푢 + ρ + 푣 + ρ = 0 휕푡 휕푡 휕푡 휕푥 휕푥 휕푦 휕푦 휕(ρ푢) 휕푢 + 푢 + ρ푢 휕푥 휕푥 휕(ρ푣) 휕푢 + 푢 + ρ푣 휕ρ 휕ρ 휕ρ 휕푢 휕푣 휕푦 휕푦 + 푢 + 푣 + ρ + = 0 휕푡 휕푥 휕푦 휕푥 휕푦 휕ρ 휕(ρ푢) 휕(ρ푣) 휕푢 휕푢 휕푢 LHS = + + 푢 + ρ + 푢 + 푣 휕푡 휕푥 휕푦 휕푡 휕푥 휕푦 Dρ 휕푢 휕푣 =0 by mass conservation =D푢ΤD푡 + ρ + = 0 D푡 휕푥 휕푦 D푢 휕푝 ρ = − + μ∇2푢 D푡 휕푥 (b) Define carefully what is meant by the statement that a flow is incompressible. To what does the continuity equation reduce in incompressible flow? Incompressible: flow-induced changes to pressure (or temperature) do not cause significant changes in density Dρ = 0 D푡 Dρ 휕푢 휕푣 + ρ + = 0 D푡 휕푥 휕푦 휕푢 휕푣 + = 0 휕푥 휕푦 휕ρ 휕 휕 2-d continuity: + (ρ푢) + (ρ푣) = 0 휕푡 휕푥 휕푦 휕 휕 휕 휕푝 휕 휕 2-d x-momentum: (ρ푢) + (ρ푢푢) + (ρ푣푢) = − + μ∇2푢 ∇2≡ + 휕푡 휕푥 휕푦 휕푥 휕푥2 휕푦2 (c) Write down conservative forms of the 3-d equations for mass and x-momentum. 휕ρ 휕 휕 휕 3-d continuity: + (ρ푢) + (ρ푣) + (ρ푤) = 0 휕푡 휕푥 휕푦 휕푧 휕 휕 휕 휕 휕푝 휕 휕 휕 3-d x-momentum: (ρ푢) + (ρ푢푢) + (ρ푣푢) + (ρ푤푢) = − + μ∇2푢 ∇2≡ + + 휕푡 휕푥 휕푦 휕푧 휕푥 휕푥2 휕푦2 휕푧2 (d) Write down the 푧-momentum equation, including the gravitational force. 휕 휕 휕 휕 휕푝 3-d z-momentum: ρ푤 + ρ푢푤 + ρ푣푤 + ρ푤푤 = − − ρ + μ∇2푤 휕푡 휕푥 휕푦 휕푧 휕푧 (e) Show that, for constant-density flows, pressure and gravity can be combined in the momentum equations via the piezometric pressure 푝 + ρ푧.
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    Continuity Equation in Pressure Coordinates Here we will derive the continuity equation from the principle that mass is conserved for a parcel following the fluid motion (i.e., there is no flow across the boundaries of the parcel). This implies that δxδyδp δM = ρ δV = ρ δxδyδz = − g is conserved following the fluid motion: 1 d(δM ) = 0 δM dt 1 d()δM = 0 δM dt g d ⎛ δxδyδp ⎞ ⎜ ⎟ = 0 δxδyδp dt ⎝ g ⎠ 1 ⎛ d(δp) d(δy) d(δx)⎞ ⎜δxδy +δxδp +δyδp ⎟ = 0 δxδyδp ⎝ dt dt dt ⎠ 1 ⎛ dp ⎞ 1 ⎛ dy ⎞ 1 ⎛ dx ⎞ δ ⎜ ⎟ + δ ⎜ ⎟ + δ ⎜ ⎟ = 0 δp ⎝ dt ⎠ δy ⎝ dt ⎠ δx ⎝ dt ⎠ Taking the limit as δx, δy, δp → 0, ∂u ∂v ∂ω Continuity equation + + = 0 in pressure ∂x ∂y ∂p coordinates 1 Determining Vertical Velocities • Typical large-scale vertical motions in the atmosphere are of the order of 0. 01-01m/s0.1 m/s. • Such motions are very difficult, if not impossible, to measure directly. Typical observational errors for wind measurements are ~1 m/s. • Quantitative estimates of vertical velocity must be inferred from quantities that can be directly measured with sufficient accuracy. Vertical Velocity in P-Coordinates The equivalent of the vertical velocity in p-coordinates is: dp ∂p r ∂p ω = = +V ⋅∇p + w dt ∂t ∂z Based on a scaling of the three terms on the r.h.s., the last term is at least an order of magnitude larger than the other two. Making the hydrostatic approximation yields ∂p ω ≈ w = −ρgw ∂z Typical large-scale values: for w, 0.01 m/s = 1 cm/s for ω, 0.1 Pa/s = 1 μbar/s 2 The Kinematic Method By integrating the continuity equation in (x,y,p) coordinates, ω can be obtained from the mean divergence in a layer: ⎛ ∂u ∂v ⎞ ∂ω ⎜ + ⎟ + = 0 continuity equation in (x,y,p) coordinates ⎝ ∂x ∂y ⎠ p ∂p p2 p2 ⎛ ∂u ∂v ⎞ ∂ω = − ⎜ + ⎟ ∂p rearrange and integrate over the layer ∫p ∫ ⎜ ⎟ 1 ∂x ∂y p1⎝ ⎠ p ⎛ ∂u ∂v ⎞ ω(p )−ω(p ) = (p − p )⎜ + ⎟ overbar denotes pressure- 2 1 1 2 ⎜ ⎟ weighted vertical average ⎝ ∂x ∂y ⎠ p To determine vertical motion at a pressure level p2, assume that p1 = surface pressure and there is no vertical motion at the surface.
  • Lattice Boltzmann Modeling for Shallow Water Equations Using High

    Lattice Boltzmann Modeling for Shallow Water Equations Using High

    Louisiana State University LSU Digital Commons LSU Doctoral Dissertations Graduate School 2010 Lattice Boltzmann modeling for shallow water equations using high performance computing Kevin Tubbs Louisiana State University and Agricultural and Mechanical College, [email protected] Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_dissertations Part of the Engineering Science and Materials Commons Recommended Citation Tubbs, Kevin, "Lattice Boltzmann modeling for shallow water equations using high performance computing" (2010). LSU Doctoral Dissertations. 34. https://digitalcommons.lsu.edu/gradschool_dissertations/34 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Doctoral Dissertations by an authorized graduate school editor of LSU Digital Commons. For more information, please [email protected]. LATTICE BOLTZMANN MODELING FOR SHALLOW WATER EQUATIONS USING HIGH PERFORMANCE COMPUTING A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Interdepartmental Program in Engineering Science by Kevin Tubbs B.S. Physics , Southern University, 2001 M.S. Physics, Louisiana State University, 2004 May, 2010 To my family ii ACKNOWLEDGMENTS I want to acknowledge the love and support of my family and friends which was instrumental in completing my degree. I would like to especially thank my parents, John and Veronica Tubbs and my siblings Kanika Tubbs and Keosha Tubbs. I dedicate this dissertation in loving memory of my brother Kendrick Tubbs and my grandmother Gertrude Nicholas. I would also like to thank Dr.
  • Chapter 3 Newtonian Fluids

    Chapter 3 Newtonian Fluids

    CM4650 Chapter 3 Newtonian Fluid 2/5/2018 Mechanics Chapter 3: Newtonian Fluids CM4650 Polymer Rheology Michigan Tech Navier-Stokes Equation v vv p 2 v g t 1 © Faith A. Morrison, Michigan Tech U. Chapter 3: Newtonian Fluid Mechanics TWO GOALS •Derive governing equations (mass and momentum balances •Solve governing equations for velocity and stress fields QUICK START V W x First, before we get deep into 2 v (x ) H derivation, let’s do a Navier-Stokes 1 2 x1 problem to get you started in the x3 mechanics of this type of problem solving. 2 © Faith A. Morrison, Michigan Tech U. 1 CM4650 Chapter 3 Newtonian Fluid 2/5/2018 Mechanics EXAMPLE: Drag flow between infinite parallel plates •Newtonian •steady state •incompressible fluid •very wide, long V •uniform pressure W x2 v1(x2) H x1 x3 3 EXAMPLE: Poiseuille flow between infinite parallel plates •Newtonian •steady state •Incompressible fluid •infinitely wide, long W x2 2H x1 x3 v (x ) x1=0 1 2 x1=L p=Po p=PL 4 2 CM4650 Chapter 3 Newtonian Fluid 2/5/2018 Mechanics Engineering Quantities of In more complex flows, we can use Interest general expressions that work in all cases. (any flow) volumetric ⋅ flow rate ∬ ⋅ | average 〈 〉 velocity ∬ Using the general formulas will Here, is the outwardly pointing unit normal help prevent errors. of ; it points in the direction “through” 5 © Faith A. Morrison, Michigan Tech U. The stress tensor was Total stress tensor, Π: invented to make the calculation of fluid stress easier. Π ≡ b (any flow, small surface) dS nˆ Force on the S ⋅ Π surface V (using the stress convention of Understanding Rheology) Here, is the outwardly pointing unit normal of ; it points in the direction “through” 6 © Faith A.
  • 1 David Apsley 3. APPROXIMATIONS and SIMPLIFIED EQUATIONS

    1 David Apsley 3. APPROXIMATIONS and SIMPLIFIED EQUATIONS

    3. APPROXIMATIONS AND SIMPLIFIED EQUATIONS SPRING 2021 3.1 Steady-state vs time-dependent flow 3.2 Two-dimensional vs three-dimensional flow 3.3 Incompressible vs compressible flow 3.4 Inviscid vs viscous flow 3.5 Hydrostatic vs non-hydrostatic flow 3.6 Boussinesq approximation for density 3.7 Depth-averaged (shallow-water) equations 3.8 Reynolds-averaged equations (turbulent flow) Examples Fluid dynamics is governed by equations for mass, momentum and energy. The momentum equation for a viscous fluid is called the Navier-Stokes equation; it is based upon: • continuum mechanics; • the momentum principle; • shear stress proportional to velocity gradient. A fluid for which the last is true is called a Newtonian fluid. This is the case for most fluids in engineering. However, there are important non-Newtonian fluids; e.g. mud, cement, blood, paint, polymer solutions. CFD is very useful for these, as their governing equations are usually impossible to solve analytically. The full equations are time-dependent, 3-dimensional, viscous, compressible, non-linear and highly coupled. However, in most cases it is possible to simplify analysis by adopting a reduced equation set. Some common approximations are listed below. Reduction of dimension: • steady-state; • two-dimensional. Neglect of some fluid property: • incompressible; • inviscid. Simplified forces: • hydrostatic; • Boussinesq approximation for density. Approximations based upon averaging: • depth-averaging (shallow-water equations); • Reynolds-averaging (turbulent flows). The consequences of these approximations are examined in the following sections. CFD 3 – 1 David Apsley 3.1 Steady-State vs Time-Dependent Flow Many flows are naturally time-dependent. Examples include waves, tides, turbines, reciprocating pumps and internal combustion engines.
  • CONTINUITY EQUATION  Another Principle on Which We Can Derive a New Equation Is the Conservation of Mass

    CONTINUITY EQUATION  Another Principle on Which We Can Derive a New Equation Is the Conservation of Mass

    ESCI 342 – Atmospheric Dynamics I Lesson 7 – The Continuity and Additional Equations Suggested Reading: Martin, Chapter 3 THE SYSTEM OF EQUATIONS IS INCOMPLETE The momentum equations in component form comprise a system of three equations with 4 unknown quantities (u, v, p, and ). Du1 p fv (1) Dt x Dv1 p fu (2) Dt y p g (3) z They are not a closed set, because there are four dependent variables (u, v, p, and ), but only three equations. We need to come up with some more equations in order to close the set. DERIVATION OF THE CONTINUITY EQUATION Another principle on which we can derive a new equation is the conservation of mass. The equation derived from this principle is called the mass continuity equation, or simply the continuity equation. Imagine a cube at a fixed point in space. The net change in mass contained within the cube is found by adding up the mass fluxes entering and leaving through each face of the cube.1 The mass flux across a face of the cube normal to the x-axis is given by u. Referring to the picture below, these fluxes will lead to a rate of change in mass within the cube given by m u yz u yz (4) t x xx 1 A flux is a quantity per unit area per unit time. Mass flux is therefore the rate at which mass moves across a unit area, and would have units of kg s1 m2. The mass in the cube can be written in terms of the density as m xyz so that m x y z .
  • On the Modeling and Design of Zero-Net Mass Flux Actuators

    On the Modeling and Design of Zero-Net Mass Flux Actuators

    ON THE MODELING AND DESIGN OF ZERO-NET MASS FLUX ACTUATORS By QUENTIN GALLAS A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005 Copyright 2005 by Quentin Gallas Pour ma famille et mes amis, d’ici et de là-bas… (To my family and friends, from here and over there…) ACKNOWLEDGMENTS Financial support for the research project was provided by a NASA-Langley Research Center Grant and an AFOSR grant. First, I would like to thank my advisor, Dr. Louis N. Cattafesta. His continual guidance and support gave me the motivation and encouragement that made this work possible. I would also like to express my gratitude especially to Dr. Mark Sheplak, and to the other members of my committee (Dr. Bruce Carroll, Dr. Bhavani Sankar, and Dr. Toshikazu Nishida) for advising and guiding me with various aspects of this project. I thank the members of the Interdisciplinary Microsystems group and of the Mechanical and Aerospace Engineering department (particularly fellow student Ryan Holman) for their help with my research and their friendship. I thank everyone who contributed in a small but significant way to this work. I also thank Dr. Rajat Mittal (George Washington University) and his student Reni Raju, who greatly helped me with the computational part of this work. Finally, special thanks go to my family and friends, from the States and from France, for always encouraging me to pursue my interests and for making that pursuit possible. iv TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................................................................................