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16.121 ANALYTICAL SUBSONIC AERODYNAMICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGYFundamentals of Fluid Mechanics1 F UNDAMENTALS OF F LUID M ECHANICS1.1 A SSUMPTIONS1. Fluid is a continuum2. Fluid is inviscid3. Fluid is adiabatic4. Fluid is a perfect gas5. Fluid is a constant-density ﬂuid6. Discontinuities (shocks, waves, vortex sheets) are treated as separate and serve as boundariesfor continuous portions of the ﬂow1.2 N OTATION0p pressure (static)ρ densityT temperature (absolute)Q velocity vector of ﬂuid particlesQ U i V j WkV control volume0S surface surrounding Vσ impermeable bodyn normal directed into the ﬂuidR gas constantF body force per unit massF ΩΩ potential of the force ﬁeldGravity ﬁeld: F g k; Ω g zph enthalpy per unit mass; h e ρc p speciﬁc heat at constant pressurec v speciﬁc heat at constant volumeγ c p /c ve internal energy per unit masss entropy per unit mass01.3 C ONTINUITY E QUATION ρ (ρQ) 0 tDρ ρ Q 0DtÑÓ ρρ(Qn)d s 0 0dV 0 V 0 tS 0 VÑ hi ρ (ρQ) dV 0 0V 0 t1

1.4 C ONSERVATION OF M OMENTUMXÑFi iV0DQ p F DtρÓ (ρQ)dV 0 ρQ(Qn)d s 0 tS 0 V1.5 C ONSERVATION OF T HERMODYNAMIC E NERGYD hQ2 i · (pQ)e F ·QDt2ρD hQ 2 i pρh ρF · QDt2 t1.6 E QUATION OF S TATEp RρT(thermally perfect gas)c p , c v constants (calorically perfect gas)2 P RESSURE DISTRIBUTION AND COMPRESSIBILITY2.1 A SSUMPTIONS1. Steady ﬂow2. Inviscid ﬂuid3. No discontinuities (shocks)4. Perfect gas5. One-dimensional motion6. Adiabatic ﬂow7. F 08. Isentropic2.2 N OTATION( )0 stagnation conditions, Q 0( ) free stream conditions, Q u c u c( ) conditions on body surface (airfoil)Q u 0 i u 0 j ωku 0 u γu2

2.3 E NERGY E QUATIONSh e pρih1d h Q2 02(Heat content plus kinetic energy is constant)2.4 P ERFECT G AS R ELATIONSp ρRTpV RTV 1ρCan show, without effort:ρV γ constant³ 1 γp constantρpa 2 γ , a speed of soundρr³ Q 2c p T0 Th³ p γ 1 iT iγ T0 1 T0 T T0 1 T0p0nh³ p γ 1 io 1γ2Q 2c p T0 1 p0h2.5 M ACH N UMBERM2 M2 22c p (T0 T ) 2c p (T0 T )Q p2aγRTγρ2c p³T0 1 2 ³ T0 1(γ 1) Tγ(c p c v ) TT0 hγ 1 2i 1 M β(γ, M )T2γγp 0 ³ T0 γ 1 β γ 1pT³1ρ0T0 γ 1 1 β γ 1ρT3

2.6 OTHER USEFUL FORMS , EXPRESSIONSQ 2 2c p (T0 T )p0 γRT0ρ0Q 2 2c p ³T 2 ³T 1 1 γRT0γ 1T0a 02³ Tγ 1 Q 2 1 2 a0T0γhpγ 1 ³ Q 2 i γ 1 1 2 a0p01ρ hγ 1 ³ Q 2 i γ 1 1 ρ02 a0a 02 γa 2 a 02 γ 1 2Q22.7 P RESSURE , VELOCITY RELATIONS IN ISENTROPIC FLOWWith some effort, one may show:γhpγ 1 2 ³Q 2 i γ 1 1 M 1 22u p Expanding the right-hand side:pγ³Q2 2 γ ³Q 2 2 4 γ(2 γ) ³Q 2 3 6 γ(2 γ)(3 2γ) ³Q 2 4 8 1 1 2 M 1 2 M 1 2 M 1 2 M .2u 8u 48u 384u p Obtain an expression forcp p p 122 ρ u LetQ u γV,γV¿1U Find c p and discuss its limitations.3 S IMILARITY OF FLOWS3.1 R EQUIREMENTS FOR SIMILARITY OF FLOWS1. Similarity in boundary geometryBoundary of one ﬂow can be made to coincide with that of another if its linear dimensions aremultiplied by a constant2. Dynamic constraintDependent variables of one ﬂow are proportional to those of another at the correspondingpoints.4

Example Problem - IllustrationConsider the dynamics of an incompressible ﬂuid ﬂow with constant.Equation of incompressibility:D p ρ ρ 0 uiDt t x iEquation of continuity: u i 0 x iIntroduce dimensionless variables:0ui ui

Fundamentals of Fluid Mechanics. 1 F. UNDAMENTALS OF . F. LUID . M. ECHANICS . 1.1 A. SSUMPTIONS . 1. Fluid is a continuum 2. Fluid is inviscid 3. Fluid is adiabatic 4. Fluid is a perfect gas 5. Fluid is a constant-density ﬂuid 6. Discontinuities (shocks, waves, vortex sheets) are treated as separate and serve as boundaries for continuous .

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