information [5]). Hereafter, the term direct numerical simulation (DNS) refers to the numerical integration of the Direct numerical simulation of a dilute suspension of finite-volume spheres requires computation of the time-varying fluid field exact equations of motion for the fluid and particle phases and updating the particle momenta and positions, taking into acwith essentially no modeling assumptions. count the effects due to particle-particle collisions. Collision calcula-For relatively dilute suspensions (i.e., volume fractions tions are inherently an order N 2 p operation, where N p is the number below 0.1%), there have been a number of investigations of particles in the system. Typical simulations contain 10 5 -10 6 parti-(see, for example, [6-9]) that neglect the finite volume of cles making the brute force computation of collisions prohibitively expensive. An alternative algorithm, based on molecular-dynamicthe particles, and rather treat the particles as point masses. simulation strategies, is proposed in this paper. A second consider-In addition, we include the Lagrangian particle studies of ation in simulating a finite-volume particle suspension is how the Yeung and Pope [10] and more recently Yeung [11] in this particle forces should be coupled back into the fluid calculation (socategory since the algorithms are essentially the same. The called reverse coupling). Careful consideration of the energy budget for the particle and fluid phases indicates that interpolation schemes critical numerical question in these studies is the interpolafor forward and reverse coupling must be symmetric in order to tion of the grid velocities to the particle positions. As shown ensure proper behavior of the overall energy balance. Asymmetric by Yeung and Pope [10] and subsequently by Balachandar interpolation schemes will lead to errors of one plus the error of and Maxey [12], the need for a high-order interpolation the least accurate interpolation method per iteration. Of course,
(third order and higher) is essential for accurate represenglobal errors may be much larger due to a cumulative effect of the systematic deviation.