Euler Solution Using Adaptive Cartesian Grid With A Gridless Boundary Treatment

A quadtree-based adaptive Cartesian grid generator and flow solver were developed. The grid adaptation based on pressure or density gradient was performed and a gridless method based on the least-square fashion was used to treat the wall surface boundary condition, which is generally difficult to be handled for the common Cartesian grid. First, to validate the technique of grid adaptation, the benchmarks over a forward-facing step and double Mach reflection were computed. Second, the flows over the NACA 0012 airfoil and a two-element airfoil were calculated to validate the developed gridless method. The computational results indicate the developed method is reasonable for complex flows.

Acoustic Calculation for Supersonic Turbulent Boundary Layer Flow

An approach which combines direct numerical simulation (DNS) with the Lighthill acoustic analogy theory is used to study the potential noise sources during the transition process of a Mach 2.25 flat plate boundary layer. The quadrupole sound sources due to the flow fluctuations and the dipole sound sources due to the fluctuating surface stress are obtained. Numerical results suggest that formation of the high shear layers leads to a dramatic amplification of amplitude of the fluctuating quadrupole sound sources. Compared with the quadrupole sound source, the energy of dipole sound source is concentrated in the relatively low frequency range.

Direct numerical simulation of hypersonic boundary layer transition over a blunt cone with a small angle of attack

The direct numerical simulation of boundary layer transition over a 5° half-cone-angle blunt cone is performed. The free-stream Mach number is 6 and the angle of attack is 1°. Random wall blow-and-suction perturbations are used to trigger the transition. Different from the authors’ previous work [Li et al., AIAA J. 46, 2899(2008)], the whole boundary layer flow over the cone is simulated (while in the author’s previous work, only two 45° regions around the leeward and the windward sections are simulated). The transition location on the cone surface is determined through the rapid increase in skin fraction coefficient (Cf). The transition line on the cone surface shows a nonmonotonic curve and the transition is delayed in the range of 0° ≤ θ ≤ 30° (θ = 0° is the leeward section). The mechanism of the delayed transition is studied by using joint frequency spectrum analysis and linear stability theory (LST). It is shown that the growth rates of unstable waves of the second mode are suppressed in the range of 20° ≤ θ ≤ 30°, which leads to the delayed transition location. Very low frequency waves VLFWs� are found in the time series recorded just before the transition location, and the periodic times of VLFWs are about one order larger than those of ordinary Mack second mode waves. Band-pass filter is used to analyze the low frequency waves, and they are deemed as the effect of large scale nonlinear perturbations triggered by LST waves when they are strong enough.The direct numerical simulation of boundary layer transition over a 5° half-cone-angle blunt cone is performed. The free-stream Mach number is 6 and the angle of attack is 1°. Random wall blow-and-suction perturbations are used to trigger the transition. Different from the authors’ previous work [ Li et al., AIAA J. 46, 2899 (2008) ], the whole boundary layer flow over the cone is simulated (while in the author’s previous work, only two 45° regions around the leeward and the windward sections are simulated). The transition location on the cone surface is determined through the rapid increase in skin fraction coefficient (Cf). The transition line on the cone surface shows a nonmonotonic curve and the transition is delayed in the range of 20° ≤ θ ≤ 30° (θ = 0° is the leeward section). The mechanism of the delayed transition is studied by using joint frequency spectrum analysis and linear stability theory (LST). It is shown that the growth rates of unstable waves of the second mode are suppressed in the range of 20° ≤ θ ≤ 30°, which leads to the delayed transition location. Very low frequency waves (VLFWs) are found in the time series recorded just before the transition location, and the periodic times of VLFWs are about one order larger than those of ordinary Mack second mode waves. Band-pass filter is used to analyze the low frequency waves, and they are deemed as the effect of large scale nonlinear perturbations triggered by LST waves when they are strong enough.

Experimental Study Of Vortex-Induced Vibrations Of A Cylinder Near A Rigid Plane Boundary In Steady Flow

In this study, the vortex-induced vibrations of a cylinder near a rigid plane boundary in a steady flow are studied experimentally. The phenomenon of vortex-induced vibrations of the cylinder near the rigid plane boundary is reproduced in the flume. The vortex shedding frequency and mode are also measured by the methods of hot film velocimeter and hydrogen bubbles. A parametric study is carried out to investigate the influences of reduced velocity, gap-to-diameter ratio, stability parameter and mass ratio on the amplitude and frequency responses of the cylinder. Experimental results indicate: (1) the Strouhal number (St) is around 0.2 for the stationary cylinder near a plane boundary in the sub-critical flow regime; (2) with increasing gap-to-diameter ratio (e (0)/D), the amplitude ratio (A/D) gets larger but frequency ratio (f/f (n) ) has a slight variation for the case of larger values of e (0)/D (e (0)/D > 0.66 in this study); (3) there is a clear difference of amplitude and frequency responses of the cylinder between the larger gap-to-diameter ratios (e (0)/D > 0.66) and the smaller ones (e (0)/D < 0.3); (4) the vibration of the cylinder is easier to occur and the range of vibration in terms of V (r) number becomes more extensive with decrease of the stability parameter, but the frequency response is affected slightly by the stability parameter; (5) with decreasing mass ratio, the width of the lock-in ranges in terms of V (r) and the frequency ratio (f/f (n) ) become larger.

Boundary value problems for stochastic differential equations

A theory of two-point boundary value problems analogous to the theory of initial value problems for stochastic ordinary differential equations whose solutions form Markov processes is developed. The theory of initial value problems consists of three main parts: the proof that the solution process is markovian and diffusive; the construction of the Kolmogorov or Fokker-Planck equation of the process; and the proof that the transistion probability density of the process is a unique solution of the Fokker-Planck equation.

It is assumed here that the stochastic differential equation under consideration has, as an initial value problem, a diffusive markovian solution process. When a given boundary value problem for this stochastic equation almost surely has unique solutions, we show that the solution process of the boundary value problem is also a diffusive Markov process. Since a boundary value problem, unlike an initial value problem, has no preferred direction for the parameter set, we find that there are two Fokker-Planck equations, one for each direction. It is shown that the density of the solution process of the boundary value problem is the unique simultaneous solution of this pair of Fokker-Planck equations.

This theory is then applied to the problem of a vibrating string with stochastic density.

Bifurcation theory of nonlinear boundary value problems

The theory of bifurcation of solutions to two-point boundary value problems is developed for a system of nonlinear first order ordinary differential equations in which the bifurcation parameter is allowed to appear nonlinearly. An iteration method is used to establish necessary and sufficient conditions for bifurcation and to construct a unique bifurcated branch in a neighborhood of a bifurcation point which is a simple eigenvalue of the linearized problem. The problem of bifurcation at a degenerate eigenvalue of the linearized problem is reduced to that of solving a system of algebraic equations. Cases with no bifurcation and with multiple bifurcation at a degenerate eigenvalue are considered.

The iteration method employed is shown to generate approximate solutions which contain those obtained by formal perturbation theory. Thus the formal perturbation solutions are rigorously justified. A theory of continuation of a solution branch out of the neighborhood of its bifurcation point is presented. Several generalizations and extensions of the theory to other types of problems, such as systems of partial differential equations, are described.

The theory is applied to the problem of the axisymmetric buckling of thin spherical shells. Results are obtained which confirm recent numerical computations.

A singularly perturbed linear two-point boundary-value problem

We consider the following singularly perturbed linear two-point boundary-value problem:

Ly(x) ≡ Ω(ε)D_xy(x) - A(x,ε)y(x) = f(x,ε) 0≤x≤1 (1a)

By ≡ L(ε)y(0) + R(ε)y(1) = g(ε) ε → 0^+ (1b)

Here Ω(ε) is a diagonal matrix whose first m diagonal elements are 1 and last m elements are ε. Aside from reasonable continuity conditions placed on A, L, R, f, g, we assume the lower right mxm principle submatrix of A has no eigenvalues whose real part is zero. Under these assumptions a constructive technique is used to derive sufficient conditions for the existence of a unique solution of (1). These sufficient conditions are used to define when (1) is a regular problem. It is then shown that as ε → 0^+ the solution of a regular problem exists and converges on every closed subinterval of (0,1) to a solution of the reduced problem. The reduced problem consists of the differential equation obtained by formally setting ε equal to zero in (1a) and initial conditions obtained from the boundary conditions (1b). Several examples of regular problems are also considered.

A similar technique is used to derive the properties of the solution of a particular difference scheme used to approximate (1). Under restrictions on the boundary conditions (1b) it is shown that for the stepsize much larger than ε the solution of the difference scheme, when applied to a regular problem, accurately represents the solution of the reduced problem.

Furthermore, the existence of a similarity transformation which block diagonalizes a matrix is presented as well as exponential bounds on certain fundamental solution matrices associated with the problem (1).

Seismic structure above and below the core-mantle boundary

Seismic structure above and below the core-mantle boundary (CMB) has been studied through use of travel time and waveform analyses of several different seismic wave groups. Anomalous systematic trends in observables document mantle heterogeneity on both large and small scales. Analog and digital data has been utilized, and in many cases the analog data has been optically scanned and digitized prior to analysis.

Differential travel times of S - SKS are shown to be an excellent diagnostic of anomalous lower mantle shear velocity (V s) structure. Wavepath geometries beneath the central Pacific exhibit large S- SKS travel time residuals (up to 10 sec), and are consistent with a large scale 0(1000 km) slower than average V_s region (≥3%). S - SKS times for paths traversing this region exhibit smaller scale patterns and trends 0(100 km) indicating V_s perturbations on many scale lengths. These times are compared to predictions of three tomographically derived aspherical models: MDLSH of Tanimoto [1990], model SH12_WM13 of Suet al. [1992], and model SH.10c.17 of Masters et al. [1992]. Qualitative agreement between the tomographic model predictions and observations is encouraging, varying from fair to good. However, inconsistencies are present and suggest anomalies in the lower mantle of scale length smaller than the present 2000+ km scale resolution of tomographic models. 2-D wave propagation experiments show the importance of inhomogeneous raypaths when considering lateral heterogeneities in the lowermost mantle.

A dataset of waveforms and differential travel times of S, ScS, and the arrival from the D" layer, Scd, provides evidence for a laterally varying V_s velocity discontinuity at the base of the mantle. Two different localized D" regions beneath the central Pacific have been investigated. Predictions from a model having a V_s discontinuity 180 km above the CMB agree well with observations for an eastern mid-Pacific CMB region. This thickness differs from V_s discontinuity thicknesses found in other regions, such as a localized region beneath the western Pacific, which average near 280 km. The "sharpness" of the V_s jump at the top of D", i.e., the depth range over which the V_s increase occurs, is not resolved by our data, and our data can in fact may be modeled equally well by a lower mantle with the increase in V_s at the top of D" occurring over a 100 krn depth range. It is difficult at present to correlate D" thicknesses from this study to overall lower mantle heterogeneity, due to uncertainties in the 3-D models, as well as poor coverage in maps of D" discontinuity thicknesses.

P-wave velocity structure (V_p) at the base of the mantle is explored using the seismic phases SKS and SPdKS. SPdKS is formed when SKS waves at distances around 107° are incident upon the CMB with a slowness that allows for coupling with diffracted P-waves at the base of the mantle. The P-wave diffraction occurs at both the SKS entrance and exit locations of the outer core. SP_dKS arrives slightly later in time than SKS, having a wave path through the mantle and core very close to SKS. The difference time between SKS and SP_dKS strongly depends on V_p at the base of the mantle near SK Score entrance and exit points. Observations from deep focus Fiji-Tonga events recorded by North American stations, and South American events recorded by European and Eurasian stations exhibit anomalously large SP_dKS - SKS difference times. SKS and the later arriving SP_dKS phase are separated by several seconds more than predictions made by 1-D reference models, such as the global average PREM [Dziewonski and Anderson, 1981] model. Models having a pronounced low-velocity zone (5%) in V_p in the bottom 50-100 km of the mantle predict the size of the observed SP_dK S-SKS anomalies. Raypath perturbations from lower mantle V_s structure may also be contributing to the observed anomalies.

Outer core structure is investigated using the family of SmKS (m=2,3,4) seismic waves. SmKS are waves that travel as S-waves in the mantle, P-waves in the core, and reflect (m-1) times on the underside of the CMB, and are well-suited for constraining outermost core V_p structure. This is due to closeness of the mantle paths and also the shallow depth range these waves travel in the outermost core. S3KS - S2KS and S4KS - S3KS differential travel times were measured using the cross-correlation method and compared to those from reflectivity synthetics created from core models of past studies. High quality recordings from a deep focus Java Sea event which sample the outer core beneath the northern Pacific, the Arctic, and northwestern North America (spanning 1/8th of the core's surface area), have SmKS wavepaths that traverse regions where lower mantle heterogeneity is pre- dieted small, and are well-modeled by the PREM core model, with possibly a small V_p decrease (1.5%) in the outermost 50 km of the core. Such a reduction implies chemical stratification in this 50 km zone, though this model feature is not uniquely resolved. Data having wave paths through areas of known D" heterogeneity (±2% and greater), such as the source-side of SmKS lower mantle paths from Fiji-Tonga to Eurasia and Africa, exhibit systematic SmKS differential time anomalies of up to several seconds. 2-D wave propagation experiments demonstrate how large scale lower mantle velocity perturbations can explain long wavelength behavior of such anomalous SmKS times. When improperly accounted for, lower mantle heterogeneity maps directly into core structure. Raypaths departing from homogeneity play an important role in producing SmKS anomalies. The existence of outermost core heterogeneity is difficult to resolve at present due to uncertainties in global lower mantle structure. Resolving a one-dimensional chemically stratified outermost core also remains difficult due to the same uncertainties. Restricting study to higher multiples of SmKS (m=2,3,4) can help reduce the affect of mantle heterogeneity due to the closeness of the mantle legs of the wavepaths. SmKS waves are ideal in providing additional information on the details of lower mantle heterogeneity.

Slender-body hypervelocity boundary-layer instability

With novel application of optical techniques, the slender-body hypervelocity boundary-layer instability is characterized in the previously unexplored regime where thermo-chemical effects are important. Narrowband disturbances (500-3000~kHz) are measured in boundary layers with edge velocities of up to 5~km/s at two points along the generator of a 5 degree half angle cone. Experimental amplification factor spectra are presented. Linear stability and PSE analysis is performed, with fair prediction of the frequency content of the disturbances; however, the analysis over-predicts the amplification of disturbances. The results of this work have two key implications: 1) the acoustic instability is present and may be studied in a large-scale hypervelocity reflected-shock tunnel, and 2) the new data set provides a new basis on which the instability can be studied.

Identification of thermally-tagged coherent structures in the zero pressure gradient turbulent boundary layer

A zero pressure gradient boundary layer over a flat plate is subjected to step changes in thermal condition at the wall, causing the formation of internal, heated layers. The resulting temperature fluctuations and their corresponding density variations are associated with turbulent coherent structures. Aero-optical distortion occurs when light passes through the boundary layer, encountering the changing index of refraction resulting from the density variations. Instantaneous measurements of streamwise velocity, temperature and the optical deflection angle experienced by a laser traversing the boundary layer are made using hot and cold wires and a Malley probe, respectively. Correlations of the deflection angle with the temperature and velocity records suggest that the dominant contribution to the deflection angle comes from thermally-tagged structures in the outer boundary layer with a convective velocity of approximately 0.8U∞. An examination of instantaneous temperature and velocity and their temporal gradients conditionally averaged around significant optical deflections shows behavior consistent with the passage of a heated vortex. Strong deflections are associated with strong negative temperature gradients, and strong positive velocity gradients where the sign of the streamwise velocity fluctuation changes. The power density spectrum of the optical deflections reveals associated structure size to be on the order of the boundary layer thickness. A comparison to the temperature and velocity spectra suggests that the responsible structures are smaller vortices in the outer boundary layer as opposed to larger scale motions. Notable differences between the power density spectra of the optical deflections and the temperature remain unresolved due to the low frequency response of the cold wire.