Computational Fluid Dynamics: The Finite-Volume Method
Computational Fluid Dynamics:The Finite-Volume MethodDavid Apsley
1. Introduction
What is Computational Fluid Dynamics?The use of computers andnumerical methods to solveproblems involving fluid flow
Aerodynamics
Wind Loading
Turbine Technology
Vortex Shedding
Dispersion of Pollution
Ventilation
Particle-Laden Plumes (Sea Outfalls)free surfacer a, Ua(ex302)ra 1040 kg/m3hDr 0, U0Wsz0solid bed(ex322)ra 1020 kg/m3(ex336)ra 1020 kg/m3Ua 0.026 m/s
Sediment ScourRiver bendBridge pier
Field variables:fd𝑓d𝑥2Equations: f1 xx Δ𝑓Δ𝑥 𝑓2 𝑓1𝑥2 𝑥1
Basic Principles of CFD1. Discretise space:replace field variables (r, 𝑢, 𝑣, 𝑤, 𝑝, ) by values at a finite number of nodes2. Discretise equations:continuum equations algebraic equations3. Solve:large system of simultaneous equations
Stages of a CFD Analysis Pre-processing:‒ formulate problem (geometry, equations, boundary conditions)‒ construct computational mesh Solving:‒ discretise‒ solve Post-processing:‒ analyse‒ visualise (graphs and plots)
Fluid-Flow Equations Mass:change of mass 0 Momentum:change of momentum force time Energy:change of energy work heat (Other constituents)In fluid mechanics, these are normally expressed in rate form
Form of Equations Integral (control-volume) Differential
Integral (Control-Volume) ApproachConsider the budget of any transported physical quantity in any control volumeVCHANGE IN OUTCHANGEtimeTIME DERIVATIVEof amount in 𝑉TIME DERIVATIVEof amount in 𝑉 OUT INtime CREATED CREATEDtime NET FLUXthrough boundary of 𝑉 ADVECTION DIFFUSIONthrough boundary of 𝑉 Finite-volume method for CFD SOURCEinside 𝑉SOURCEinside 𝑉
Differential Equations For Fluid Flow Derived by considering the rate of change at a point; i.e. usinginfinitesimal control volumes Discretisation gives a finite-difference method for CFD Several types:‒ fixed-point (“Eulerian”): conservative‒ moving with the flow (“Lagrangian”): non-conservative‒ derived variables; e.g. potential flow
Main Methods for CFD i,j 1Finite-difference:‒ discretise differential equations0 𝑢 𝑣 𝑥 𝑦 i-1,j𝑢𝑖 1,𝑗 𝑢𝑖 1,𝑗 𝑣𝑖,𝑗 1 𝑣𝑖,𝑗 1 2Δ𝑥2Δ𝑦i,ji,j-1Finite-volume:‒ discretise control-volume equations0 net mass outflow ρ𝑢𝐴𝑒 ρ𝑢𝐴vn𝑤 ρ𝑣𝐴𝑛 ρ𝑣𝐴𝑠uwuevs i 1,jFinite-element:‒ represent solution as a weighted sum of basis functions𝑢(x) 𝑢α 𝑆α (x)
Advantages of the Finite-Volume Method in CFD Rigorously enforces conservation Flexible in terms of:‒ geometry‒ fluid phenomena Directly relatable to physical quantities
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Examples
Example Q1Water (density 1000 kg m–3) flows at 2 m s–1 through acircular pipe of diameter 10 cm. What is the mass flux 𝐶across the surfaces 𝑆1 and 𝑆2 ?10 cmo2 m/s45S1S2
10 cmWater (density 1000 kg m–3) flows at 2 m s–1 through a circular pipe of diameter 10 cm. What isthe mass flux 𝐶 across the surfaces 𝑆1 and 𝑆2 ?o𝑆1 :mass flux (𝐶) ρ𝑢𝐴𝑆2 :the same!π 0. 12 1000 2 4 15.7 kg s 1𝐶 ρ(𝑢 cos θ)𝐴𝐶 ρ𝑢(𝐴 cos θ)In general:𝐶 ρu A2 m/s45S1S2
Example Q2A water jet strikes normal to a fixed plate as shown.Compute the force 𝐹 required to hold the plate fixed.D 10 cmu 8 m/sF
A water jet strikes normal to a fixed plate as shown. Compute the force 𝐹 required to hold theplate fixed.D 10 cmu 8 m/sforce on fluid momentum fluxout momentum fluxinmomentum flux mass flux velocity 𝐹 0 (ρ𝑈𝐴)𝑈𝐹 ρ𝑈 2 𝐴2π 0.1 1000 82 4 503 NF
Example Q3An explosion releases 2 kg of a toxic gas into a room of dimensions30 m 8 m 5 m. Assuming the room air to be well-mixed and tobe vented at a speed of 0.5 m s–1 through an aperture of area 6 m2,calculate:(a) the initial concentration of gas in ppm by mass;(b) the time taken to reach a safe concentration of 1 ppm.(Take the density of air as 1.2 kg m–3.)
An explosion releases 2 kg of a toxic gas into a room of dimensions 30 m 8 m 5 m. Assumingthe room air to be well-mixed and to be vented at a speed of 0.5 m s–1 through an aperture ofarea 6 m2, calculate:(a) the initial concentration of gas in ppm by mass;Volume V𝑉 30 8 5Concentration 1200 m3area AUmass of fluid concentration mass of toxin(ρ𝑉)ϕ0 2 kg2ϕ0 1.2 1200 1.389 10 31389 ppm
An explosion releases 2 kg of a toxic gas into a room of dimensions 30 m 8 m 5 m. Assumingthe room air to be well-mixed and to be vented at a speed of 0.5 m s–1 through an aperture ofarea 6 m2, calculate:(b) the time taken to reach a safe concentration of 1 ppm.ϕ0 1389 ppmVolume VConcentration area AU𝑉 1200 m3𝐴 6 m2𝑈 0.5 m s 1Change in amount of toxin amount in amount outRate of change of amount of toxin rate of entering rate of leavingd(ρ𝑉ϕ) 0 (ρ𝑢𝐴)ϕd𝑡dϕ𝑢𝐴 ϕ,d𝑡𝑉dϕ λϕd𝑡ϕ ϕ0 e λ𝑡ϕ ϕ0 at 𝑡 0𝑢𝐴λ 0.0025 s 1𝑉ϕ0 ϕ1 ϕ0𝑡 lnλ ϕeλ𝑡1 ln 13890.0025 2895 s 48 min
Computational Fluid Dynamics? The use of computers and numerical methods to solve problems involving fluid flow. . Basic Principles of CFD 1. Discretise space: replace fi
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1.1 What is computational fluid dynamics? 1.2 Basic principles of CFD 1.3 Stages in a CFD simulation 1.4 Fluid-flow equations 1.5 The main discretisation methods Appendices Examples 1.1 What is Computational Fluid Dynamics? Computational fluid dynamics (CFD) is the use of computers and
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