Download Computational Fluid Mechanics. Selected Papers by Alexandre Joel Chorin PDF

By Alexandre Joel Chorin

Listed below are 14 papers released through Chorin (math, UC Berkeley) among 1967 and 1 on vortex tools, the projection procedure, and the numerical resolution of difficulties in kinetic conception, combustion thought, and fuel dynamics. those paper clarify, between different subject matters, how one may well manage a discrete approximati

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I i i > i i i i where Σ » denotes summation over the ith (/-chain. Let N i be the number of points in the ith (j-chain. We have Ni ^ (N - 1)2/L > \N2/L. Therefore and ~ Σ \ΨLtf ^ 2 ( Σ ^ > 2 ) 2 + 8||G*||2. Summing over all G-chains we obtain (12) IWI2^2L Σ G-chains ( Σ ^ . ^ ν + σ,ιιοψΐΓ, \ i / where Ci = 8ZA A similar inequality can be derived in the three-dimensional case, with Ci = 12ZA The inequality (12) can now be used to prove the following theorem: 40 ALEXANDRE JOEL CHORIN THEOREM holds (13) 2.

Application to Thermal Convection. Suppose a plane layer of fluid, in the field of gravity, of thickness d and infinite lateral extent, is heated from below. The lower boundary xz = 0 is maintained at a temperature T0f the upper boundary xz = d at a temperature ΤΊ < T0. The warmer fluid at the bottom expands and tends to move upward; this motion is inhibited by the viscous stresses. g. [9]) the equations describing the possible motions are dtUi + UjdjUi = - - dip + vV2Ui - 0(1 - (T Po dtT + ujdjT = kV2T , Ί\))δ{, djUj = 0 , where T is the temperature, k the coefficient of thermal conductivity, a the co­ efficient of thermal expansion, and δ,· the components of the unit vector pointing upwards.

117, and C is a small but undetermined amplitude. In two-dimensional motion Ui = 0 and the motion does not depend on xi. We then have φ = cos ax2. The motion is periodic in x2 with period 2ir/a. The Nusselt number Nu is defined as the ratio of the total heat transfer to the heat transfer which would have occurred if no convection were present. For R* g Rc*y Nu = 1. In our dimensionless variables f2r/a Nu = — / (u3T - dzT)dx2. A similar expression holds in the three-dimensional case. When the convection is steady Nu does not depend on z 3 .

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