Rick L. Smith

Computation of aerodynamic coefficients for a flexible membrane airfoil in turbulent flow: A comparison with classical theory

In the present paper an aeroelastic model of flexible membrane wing aerodynamics which incorporates the Reynolds‐averaged Navier–Stokes equations is presented. The Reynolds stresses are prescribed by the k–ω shear‐stress transport eddy‐viscosity model recently proposed by Menter. The computed coefficients are compared with classical inviscid membrane airfoil theory and with a portion of the available experimental data for membrane wings. The results indicate that classical potential‐based membrane airfoil theory can provide a meaningful description of membrane wing aerodynamics only for a small range of incidence angles near ideal and then only for membrane airfoils with small excess length ratios. For larger excess lengths and incidence angles viscous effects dominate the aerodynamics. The agreement of the computed results with the experimental data is mixed. The current status of the available experimental data for membrane airfoils is also reviewed.

Computational model of flexible membrane wings in steady laminar flow

A computational procedure is presented that models the interaction of a two-dimensional flexible membrane wing and laminar, high-Reynolds-number fluid flow. The membrane wing model is derived by combining a spatial-coordinate-based finite difference formulation of the equilibrium statement for an elastic membrane with a pressure-based control volume formulation of the incompressible Navier-Stokes equations written in general curvilinear body-fitted coordinates. The model is applied to initially flat membrane wings of both vanishing and finite material stiffness as well as to flexible inextensible wings with excess length. Computational results are presented for Reynolds numbers between 2 x 10 3 and 10 4 . The results from the viscous-flow-based membrane wing model are compared with predictions using a potential-flow-based model as well as with experimental data for membrane wings in turbulent flow. Although the assumption of laminar flow precludes a quantitative comparison with the available experimental data, the solutions obtained capture many of the significant features of the aeroelastic interaction that are unaccounted for with a potential flow description of the fluid dynamics.

Computation of unsteady laminar flow over a flexible two‐dimensional membrane wing

Computations of the harmonically forced, unsteady viscous flow over a flexible, two‐dimensional membrane wing are presented. The aeroelastic problem is nondimensionalized and a set of six basic dimensionless parameters is derived that govern the physical problem. The computational investigation is facilitated by distinguishing three distinct classes of problems—the constant tension, elastic, and inextensible membrane problems—which are associated with limiting cases of the dimensionless parameter set. A pressure‐based method for the incompressible Navier–Stokes equations written in general time‐dependent curvilinear coordinates is adopted as the flow solver. The computations were performed at a Reynolds number of 4×103, which is near the upper limit of the laminar flow regime. The periodic appearance and collapse of recirculation zones, along with an attendant adjustment in membrane configuration, results in an aeroelastic response, which may not be characterized as a simple harmonic response at the free‐...

The role of mass conservation in pressure-based algorithms

Two numerical issues important to proper problem specification for pressure-based algorithms are investigated, including (1) well posedness of the pressure-correction equation, and (2) proper prescription of flow variables at open boundaries, particularly if inflow occurs. Lid-driven cavity flow and flow past a backward-facing step are used to help discuss the issues. It is shown that during each iteration, the explicit enforcement of global mass conservation is important even for the intermediate, nonconvergent flow field in order to maintain good convergence rates. This requirement stems from the fact that the pressure distribution is an outcome of the continuity equation. Furthermore, it seems that the global continuity constraint is often sufficient for the numerical problem for a flow with an open boundary to be well posed, regardless of whether or not inflow occurs at that boundary. Thus, in the pressure-based algorithm with a staggered grid the downstream boundary can, if necessary, pass through a ...

Addendum to “Countable algebra and set existence axioms”

Proof. (1) + (2) is given on p. 163 of [l]. (2) j (3) is trivial. (3) j (1). We shall use Lemma 3.2 of [l]. Let f, g :N +-N be given with Vi, j ((f(i) # go’)) and f, g both one-to-one. Let R, = Q[x,, : n EN] be the polynomial ring over the rationals with countably many indeterminates. Now let I be the ideal generated by all x;(‘;;t and

Incremental potential flow based membrane wing element

!tnf 1 for n EN. We claim that I exists. Given an f E R, we can write f in normal form, f*, where if xk occurs in f*, then m # f(n), g(n) for all n < k, and f = f* (mod I). Clearly f E I iff f* = 0, and l

On the ranked points of a Π1 0 set

I. Let R = Roll and assuming (3) let J be a radical ideal in R. Let Jo be the ideal in J which corresponds to J. Thus Jo is a radical ideal in R, and 1~ Jo. It follows that +(,, EJO and X,H &Jo for all HEN. Now let X={mEN:x,,,EJo} and we conclude that (1) holds.

Effective Aspects of Profinite Groups

A new quadrilateral membrane wing element is presented that combines an incremental formulation of the elastic membrane problem with a vortex lattice formulation of the thin wing aerodynamic problem. The incremental formulation leads to an explicit/implicit velocity-stepping algorithm for solving the coupled aeroelastostatic membrane wing problem. The Stein-Hedgepeth wrinkle model is adopted as the constitutive relation for the membrane, which may develop slack or wrinkled regions. The element is shown to satisfy two limiting cases of the aeroelastic problem for which there are well-known analytic solutions. The algorithm is applied to two moderate aspect ratio membrane wing configurations with the second configuration being a model of a marine sail.

Coupled computations of a flexible membrane wing and unsteady viscous flow

The rank of an element of the Cantor space is considered and ranked points that are not π 2 0 singletons are constructed. Results about Turing equivalence in terms of ranks of points are givers

Computational Fluid Dynamics with Moving Boundaries

Profinite groups are Galois groups. The effective study of infinite Galois groups was initiated by Metakides and Nerode [8] and further developed by LaRoche [5]. In this paper we study profinite groups without considering Galois extensions of fields. The Artin method of representing a finite group as a Galois group has been generalized (effectively!) by Waterhouse [14] to profinite groups. Thus, there is no loss of relevance in our approach. The fundamental notions of a co-r.e. profinite group, recursively profinite group, and the degree of a co-r.e. profinite group are defined in ?1. In this section we prove that every co-r.e. profinite group can be effectively represented as an inverse limit of finite groups. The degree invariant is shown to behave very well with respect to open subgroups and quotients. The work done in this section is basic to the rest of the paper. The commutator subgroup, the Frattini subgroup, the p-Sylow subgroups, and the center of a profinite group are essential in the study of profinite groups. It is only natural to ask if these subgroups are effective. The following question exemplifies our approach to this problem: Is the center a co-r.e. profinite group? Theorem 2 provides a general method for answering this type of question negatively. Examples 3, 4 and 5 are all applications of this theorem. Super-minimal groups are those co-r.e. profinite groups with the fewest co-r.e. subgroups possible. They are defined and studied in ?3. The only naturally occurring super-minimal groups are the p-adic integer groups which have the fewest closed subgroups possible. This is the content of Theorem 3. An application of Theorem 2 produces super-minimal groups of the form II,, Z(p). The abelian superminimal groups can be classified, but Theorem 4 shows that there are nonabelian super-minimal groups. We believe that there is a general classification of the superminimal groups. A profinite group is a compact, Hausdorff, totally disconnected topological group. Every profinite group can be represented as an inverse limit of finite groups. Galois groups are profinite groups in the Krull topology, and, conversely, every profinite group is a Galois group. If {G, 7Cn} is an inverse system, we write proj lim Gn for the inverse limit of this system. Z(n) denotes the cyclic group of order n. The p-adic integers are the inverse limit, Zp = proj lim Z(pn). If {Gn} is any