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6/26/11 Lecture 5 Conditions at a discontinuity Dimensionless parameters 1 Conservation conditions at an interface The time rate of a change of a quantity ϕ(x, t) over a given volume V, moving with velocity VI is1 � � � d ∂ϕ ϕdV = dV + ϕ(VI · n) dS dt V V ∂t S Applied to mass conservation (ϕ = ρ) we � � � d ∂ρ ρdV = dV + ρ(VI · n) dS dt V V ∂t S Using the continuity equation � � V � ∂ρ + ∇ · ρv dV = 0 ∂t d dt � V or ρdV = − � S � V ∂ρ dV + ∂t � S ρ(v · n) dS = 0 ρ(v − VI )·n dS 1 This is the multidimensional analogue of Leibnitz’s theorem from calculus; it is a kinematic relation, not a law of fluid mechanics. 2 Copyright ©2011 by Moshe Matalon. This material is not to be sold, reproduced or distributed without the prior wri@en permission of the owner, M. Matalon. 1 6/26/11 When the control volume shrinks to the surface SI � � � � � � d + − lim ρdV = − ρ+(v −VI )·n+ + ρ−(v −VI )·n− dS V→0 dt V SI � �� � =0 v − VI no mass accumulation or source/sink of mass at the interface. n+ = n − + ρ+(v · n − VI ) − ρ−(v · n − VI ) n− [ ρ(v·n−VI ) ] = 0 “pillbox” control volume VI is the velocity of the interface (control surface) VI = VI ·n [ pn + ρ(v·n−VI )v − Σ · n ] = 0 results from the momentum equation,. Similarly we can derive jump relations for energy and species concentrations. In fluid mechanics, discontinuities are allowed within the continuum framework, provided the variables across the surface of discontinuity are such as to satisfy the fundamental conservation laws, or the appropriate jump conditions. 3 If viscosity is negligible [ ρ(v·n−VI ) ] = 0 [ pn + ρv(v·n−VI ) ] = 0 � � � � ρ h + 12 v 2 (v·n−VI ) + q·n = 0 [ ρYi ((v + Vi )·n−VI ) ] = 0 The momentum jump can be decomposed into normal and tangential components, to give [ρ(v·n−VI )] = 0 [p + ρ(v·n−VI )(v·n)] = 0 [n × (v × n] = 0 Rankine-Hugoniot relations 4 Copyright ©2011 by Moshe Matalon. This material is not to be sold, reproduced or distributed without the prior wri@en permission of the owner, M. Matalon. 2 6/26/11 Non-dimensional Equations In the following the chemistry will be represented by a global one-step (irreversible) reaction νF Fuel + νO Oxidizer → Products describing the combustion of a single fuel. The reaction will be assumed of order nF , nO , with respect to the fuel/oxidizer, and an overall order n = nF + nO . The reaction rate will be assumed to obey an Arrhenius law ω=B � ρYF WF �nF� ρYO WO �nO e−E/RT with an overall activation energy E and a pre-exponential factor B (treated as constant). 5 For simplicity, we will treat µ, λ, ρDi constants (although most of the theoretical development could accommodate a temperature-dependent λ) so that ∂ρ + ∇ · ρv = 0 ∂t � � Dv ρ = −∇p + µ ∇2 v + 13 ∇(∇ · v) + ρg Dt ρ DYi − ρDi ∇2 Yi = −νi Wi ω, Dt ρcp i = F, O DT Dp − λ∇2 T = + Φ + Qω Dt Dt p = ρRT /W ω=B � ρYF WF �nF� ρYO WO �nO e−E/RT 6 Copyright ©2011 by Moshe Matalon. This material is not to be sold, reproduced or distributed without the prior wri@en permission of the owner, M. Matalon. 3 6/26/11 Characteristic values: the fresh unburned state p0 , ρ0 , T0 (satisfying p0 = ρ0 RT0 /W ) for p, ρ, T . a characteristic speed v0 to be specified the diffusion length lD ≡ λ/ρcp v0 for distances the diffusion time lD /v0 for t This choice is clearly not unique and there may be other length, time, and velocity scales that, for a given problem, could be more relevant. We will use the same variables for the dimensionless quantities; i.e. after substituting v ∗ = v/v0 say, we remove the ∗ superscript. 7 ∂ρ + ∇ · ρv = 0 ∂t ρ � � Dv 1 =− ∇p + Pr ∇2 v + 13 ∇(∇ · v) + Fr−1 ρeg Dt γM 2 DYF 2 − Le−1 F ∇ YF = −ω Dt DYO 2 ρ − Le−1 O ∇ YO = −νω Dt � � DT γ −1 Dp 2 2 ρ −∇ T = + γP rM Φ + q ω Dt γ Dt ρ p = ρT ω = D ρn YFnF YOnO e−β0 /T 8 Copyright ©2011 by Moshe Matalon. This material is not to be sold, reproduced or distributed without the prior wri@en permission of the owner, M. Matalon. 4 6/26/11 Dimensionless Parameters v0 M= � γp0 /ρ0 Pr = µcp λ Mach number Prandtl number Fr = Lei = −1 Note that Pr = ρ0 v0 lD /µ = Re where Re is the Reynolds number based on the diffusion length. The Reynolds number based on a hydrodynamic length L will be large because typically L � lD . q= Q/νF WF c p T0 heat release v02 /lD g λ/ρcp Di β0 = E/RT0 ν= ν O WO ν F WF Froude number Lewis number of species i Activation energy mass weighted stoichiometric coeff. numerator is the heat release per unit mass of fuel D= (lD /v0 ) [(ρ0 /WF )nF −1 (ρ0 /WO )nO B]−1 Damköhler number = flow time reaction time 9 Low Mach Number Approximation The propagation speed of ordinary deflagration waves is in the range 1-100 cm/s, namely much smaller than the speed of sound (in air a0 = 34, 000 cm/s). M �1 momentum equation ⇒ ∇p = 0 p = P (t) + γM 2 p̂(x, t) + · · · will all other variables expressed as v + γM 2 v̂ + · · · ρ � � Dv = −∇p̂ + Pr ∇2 v + 13 ∇(∇ · v) + Fr−1 ρeg Dt DT γ −1 dP ρ − ∇2 T = +qω Dt γ dt ρT = P (t) acoustic disturbances travel infinitely fast, and are filtered out. 10 Copyright ©2011 by Moshe Matalon. This material is not to be sold, reproduced or distributed without the prior wri@en permission of the owner, M. Matalon. 5 6/26/11 Unless P (t) is specified, we are missing an equation, since p has been replaced by two variables P and p̂. An equation in bounded problems can be obtained as follows: ρ ∂T γ −1 dP + ρv · ∇T − ∇2 T = +qω ∂t γ dt ∂ρ + ∇ · ρv = 0 /T ∂t ∂(ρT ) γ −1 dP + ∇ · (ρvT ) − ∇2 T = +qω ∂t γ dt 1 dP = −∇ · (P v − ∇T ) + qω γ dt 1 γ � V dP dV = − dt � S (P v−∇T ) · n dS + q � ωdV V dP γq = dt V on the surface S, v · n = 0 and for adiabatic conditions ∂T /∂n = 0. � ω dV V 11 The low Mach number equations are (with the ”hat” in p removed), therefore ∂ρ + ∇ · ρv = 0 ∂t � � Dv ρ = −∇p + Pr ∇2 v + 13 ∇(∇ · v) + Fr−1 ρeg Dt DYF 2 ρ − Le−1 F ∇ YF = −ω Dt DYO 2 ρ − Le−1 O ∇ YO = −νω Dt DT γ − 1 dP ρ − ∇2 T = +qω Dt γ dt ρT = 1 and when the underlying pressure does not change in time, P = 1. Unless otherwise indicated, we will be mostly concerned in the following with this set of equations. 12 Copyright ©2011 by Moshe Matalon. This material is not to be sold, reproduced or distributed without the prior wri@en permission of the owner, M. Matalon. 6 6/26/11 Coupling Functions DYF 2 − Le−1 F ∇ YF = −ω Dt DYO 2 ρ − Le−1 O ∇ YO = −νω Dt DT ρ − ∇2 T = q ω Dt ρ For unity Lewis numbers the operator on the left hand side of these three equations is the same. The combinations HF = T + qYF and HO = T + q YO /ν (and hence YF − YO /ν) satisfy reaction-free equations ρ DHi − ∇ 2 Hi = 0 Dt leaving only one equation with the highly nonlinear reaction rate term. This is a great simplification, but as we shall see, small variations of the Lewis numbers from one produce instabilities and nontrivial consequences. 13 The constant-density approximation ρT = 1 ∂ρ + ∇ · ρv = 0 ∂t � � ∂v ρ( + (v · ∇)v) = −∇p + Pr ∇2 v + 13 ∇(∇ · v) + Fr−1 ρeg ∂t solve to determine v ∂YF 2 + v · ∇YF ) − Le−1 F ∇ YF = −ω ∂t ∂YO 2 ρ( + v · ∇YO ) − Le−1 O ∇ YO = −νω ∂t ∂T ρ( + v · T ) − ∇2 T = q ω ∂t ρ( with the given v solve for T, YF , YO . often referred to as the diffusive-thermal model 14 Copyright ©2011 by Moshe Matalon. This material is not to be sold, reproduced or distributed without the prior wri@en permission of the owner, M. Matalon. 7