Prediction of natural convection flow using network model and numerical simulations inside enclosure with distributed solid blocks

Steady state natural convection of a fluid with Pr ≈ 1 within a square enclosure containing uniformly distributed, conducting square solid blocks is investigated. The side walls are subjected to differential heating, while the top and bottom ones are kept adiabatic. The natural convection flow is predicted employing the nondimensional volumetric flow rate (Q_max^* Q_{\max }^{*} ) by using a network model and also using numerical simulations. For identical solid and fluid thermal conductivities (i.e. k _s  = k _f ), a parametric study of the effect of number of blocks (N ²), gap size (δ) and enclosure Rayleigh number (Ra) on Q_max^* Q_{\max }^{*} is performed using the two approaches. Network model predictions are observed to agree well with that from the simulations until Raδ³ ~ 12. Considering the enclosure with blocks as a porous medium, for a fixed enclosure Ra number, increasing the number of blocks for a fixed volumetric porosity leads to a decrease in enclosure permeability, which in turn reduces the flow rate. When the number of blocks is fixed, and for a given Ra number, the flow rate increases as the porosity increases by widening the gap between the blocks.     

Asymptotics of the Fast Diffusion Equation via Entropy Estimates

We consider non-negative solutions of the fast diffusion equation u _t  = Δ u ^m with m ∈ (0, 1) in the Euclidean space , d ≧ 3, and study the asymptotic behavior of a natural class of solutions in the limit corresponding to t → ∞ for m ≧ m _c  = (d − 2)/d, or as t approaches the extinction time when m < m _c . For a class of initial data, we prove that the solution converges with a polynomial rate to a self-similar solution, for t large enough if m ≧ m _c , or close enough to the extinction time if m < m _c . Such results are new in the range m ≦ m _c where previous approaches fail. In the range m _c  < m < 1, we improve on known results.     

Secondary flow patterns and mixing in laminar pulsating flow through a curved pipe

Mixing by secondary flow is studied by particle image velocimetry (PIV) in a developing laminar pulsating flow through a circular curved pipe. The pipe curvature ratio is η = r ₀/r _c  = 0.09, and the curvature angle is 90°. Different secondary flow patterns are formed during an oscillation period due to competition among the centrifugal, inertial, and viscous forces. These different secondary-flow structures lead to different transverse-mixing schemes in the flow. Here, transverse mixing enhancement is investigated by imposing different pulsating conditions (Dean number, velocity ratio, and frequency parameter); favorable pulsating conditions for mixing are introduced. To obviate light-refraction effects during PIV measurements, a T-shaped structure is installed downstream of the curved pipe. Experiments are carried out for the Reynolds numbers range 420 ≤ Re_st ≤ 1,000 (Dean numbers 126.6 ≤ Dn ≤ 301.5) corresponding to non-oscillating flow, velocity component ratios 1 ≤ (β = U _max,osc/U _m,st) ≤ 4 (the ratio of velocity amplitude of oscillations to the mean velocity without oscillations), and frequency parameters 8.37 < (α = r ₀(ω/ν)^0.5) < 24.5, where α² is the ratio of viscous diffusion time over the pipe radius to the characteristic oscillation time. The variations in cross-sectional average values of absolute axial vorticity (|ζ|) and transverse strain rate (|ε|) are analyzed in order to quantify mixing. The effects of each parameter (Re_st, β, and α) on transverse mixing are discussed by comparing the dimensionless vorticities (|ζ _P |/|ζ _S |) and dimensionless transverse strain rates (|ε _P |/|ε _S |) during a complete oscillation period.     

Heat and mass transfer characteristics of the self-similar boundary-layer flows induced by continuous surfaces stretched with rapidly decreasing velocities

The mechanical and thermal characteristics of the self-similar boundary-layer flows induced by continuous surfaces stretched with rapidly decreasing power-law velocities U _w∝x ^m , m<?1 are considered. Comparing to the well studied cases of the increasing stretching velocities (m>0) several new features of basic significance have been found. Thus: (i) for m<?1 the boundary layer equations admit self-similar solutions only if a lateral suction is applied; (ii) the dimensionless suction velocity f _w<0 must be strong enough, i.e. f _w<f _w,max(m) where f _w,max(m) depends on m so that its absolute maximum max (f _w,max(m))=?2.279 is reached for m→?∞, while for m→?1, f _w,max(m)→?∞; (iii) the case {m→?∞, f _w,max(m)=?2.279} of the flow boundary value problem is isomorphic to the stretching problems with exponentially decreasing velocities U _w∝e ^ax with arbitrary a<0; (iv) for any fixed m<?1 and f _w<f _w,max(m) the flow problem admits a non-denumerable infinity of multiple solutions corresponding to the values of the dimensionless skin friction f ^″(0)≡s belonging to a finite interval s∈ [s _min(f _w,m), s _max(f _w,m)]; (v) the solution is only unique for f _w=f _w,max(m) where s=s _min(f _w,m)= s _max(f _w,m) holds; (vi) to every one of the multiple solutions of the flow problem there corresponds a unique solution of the heat transfer problem with a wall temperature distribution T _w?T _∞∝x ⁿ and a well defined and distinct value of the dimensionless wall temperature gradient ?^′(0), except for the cases n=(|m|?1)/2 where ?^′(0) has the same value ?^′(0)=Pr·f _w for the whole class of flow solutions with s∈[s _min(f _w,m), s _max(f _w,m)]; (vii) for f _w→?∞ one obtains the `asymptotic suction profiles' corresponding to s=s _min(f _w,m)?f _w and ?^′(0)?Pr·f _w in an explicit analytic form. The paper includes several examples which illustrate the dependence of the heat and fluid flows induced by surfaces stretching with rapidly decreasing velocities on the physical parameters f _w, m, n and Pr.     

Measurement of flow-induced pressures on the surface of a model in a flow visualization water tunnel

Flow-induced pressures were measured on the surface of a delta wing in a flow visualization water tunnel, and at the same time the vortical flow over the wing was visualized using dye. For a free-stream velocity of 0.1 m/s used in the experiments, the pressures were small (less than 15 Pa). This is believed to be the first time that flow-induced pressures have been measured in a water tunnel at such a low velocity. Measured pressure distributions can now be properly interpreted in terms of observed detailed flow patterns, without having to undertake complementary wind tunnel experiments to measure model loading.Symbols   c   centreline chord of the delta wing (c=300.0 mm) - C _p   pressure coefficient on the surface of the delta wing - f   frequency (Hz) - p _ref   static pressure measured by the reference static-pressure probe (Pa) - P _ref   total pressure measured by the reference total-pressure probe (Pa) - p _wing   flow-induced pressure at a pressure tapping on the delta wing (Pa) - R   Reynolds number for the delta wing in water (R=Uc/v _w) - s   local semi-span of the delta wing (mm) - U   free-stream velocity in the test section of the water tunnel (m/s) - x   chordwise distance from the apex of the delta wing (mm) - y   spanwise distance from the centreline chord of the delta wing (mm) -   angle of attack of the delta wing (deg) -   angle of sideslip of the delta wing (deg) - v _w   kinematic viscosity of water (m²/s) - _a   density of air (kg/m³) - _w   density of water (kg/m³)

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Roughness induced forced convective laminar-transitional micropipe flow: energy and exergy analysis

Variable fluid property continuity, Navier–Stokes and energy equations are solved for roughness induced forced convective laminar-transitional flow in a micropipe. Influences of Reynolds number, heat flux and surface roughness, on the momentum-energy transport mechanisms and second-law of thermodynamics, are investigated for the ranges of Re = 1–2,000, Q = 5–100 W/m² and ε = 1–50 μm. Numerical investigations put forward that surface roughness accelerates transition with flatter velocity profiles and increased intermittency values (γ); such that a high roughness of ε = 50 μm resulted in transitional character at Re _tra = 450 with γ = 0.136. Normalized friction coefficient (C _f^*) values showed augmentation with Re, as the evaluated C _f^* are 1.006, 1.028 and 1.088 for Re = 100, 500 and 1,500, respectively, at ε = 1 μm, the corresponding values rise to C _f^* = 1.021, 1.116 and 1.350 at ε = 50 μm. Heat transfer rates are also recorded to rise with Re and ε; moreover the growing influence of ε on Nusselt number with Re is determined by the Nu _ε=50 μm/Nu _ε=1 μm ratios of 1.086, 1.168 and 1.259 at Re = 500, 1,000 and 1,500. Thermal volumetric entropy generation values decrease with Re and ε in heating; however the contrary is recorded for frictional volumetric entropy generation data, where the augmentations in are more considerable when compared with the decrease rates of      

Average and extremal properties of heat transfer and shear stress on a wall surface in Rayleigh–Bénard convection

In this study surface-averaged and extremal properties of heat transfer and shear stress on the upper wall surface of Rayleigh–Bénard convection are numerically examined. The Prandtl number was raised up to 10³, and the Rayleigh number was changed between 10⁴ and 10⁷. As a result, average Nusselt number Nu and shear rate τ/Pr depends on Pr, Ra, and the entire numerical results are distributed between two correlation equations corresponding to small and large Pr. The small and large Pr equations are closely related to steady and unsteady flow regimes, respectively. Nevertheless, a single relation τ/Pr ~ Nu ^3.0 exists to explain the entire results. Similarly the change of local maximal properties Nu _max and τ _max/Pr depends on Pr, Ra, and these values are also distributed between two correlation equations corresponding to small and large Pr cases. Despite such complicated dependence we can obtain a correlation equation as a form of τ _max/Pr ~ Nu _max^2.6, which has not been obtained theoretically.     

The effects of span position of winglet vortex generator on local heat/mass transfer over a three-row flat tube bank fin

The naphthalene sublimation method was used to study the effects of span position of vortex generators (VGs) on local heat transfer on three-row flat tube bank fin. A dimensionless factor of the larger the better characteristics, JF, is used to screen the optimum span position of VGs. In order to get JF, the local heat transfer coefficient obtained in experiments and numerical method are used to obtain the heat transferred from the fin. A new parameter, named as staggered ratio, is introduced to consider the interactions of vortices generated by partial or full periodically staggered arrangement of VGs. The present results reveal that: VGs should be mounted as near as possible to the tube wall; the vortices generated by the upstream VGs converge at wake region of flat tube; the interactions of vortices with counter rotating direction do not effect Nusselt number (Nu) greatly on fin surface mounted with VGs, but reduce Nu greatly on the other fin surface; the real staggered ratio should include the effect of flow convergence; with increasing real staggered ratio, these interactions are intensified, and heat transfer performance decreases; for average Nu and friction factor (f), the effects of interactions of vortices are not significant, f has slightly smaller value when real staggered ratio is about 0.6 than that when VGs are in no staggered arrangement. A cross section area of flow passage [m²] - A _mim minimum cross section area of flow passage [m²] - a width of flat tube [m] - b length of flat tube [m] - B _pT lateral pitch of flat tube: B _pT = S ₁/T _p - d _h hydraulic diameter of flow channel [m] - D _naph diffusion of naphthalene [m²/s] - f friction factor: f = pd _h/(Lu ² _max/2) - h mass transfer coefficient [m/s] - H height of winglet type vortex generators [m] - j Colburn factor [–] - JF a dimensionless ratio, defined in Eq. (23) [–] - L streamwise length of fin [m] - L _PVG longitudinal pitch of vortex generators divided by fin spacing: L _pVG = l _VG/T _p - l _VG pitch of in-line vortex generators [m] - m mass [kg] - m mass sublimation rate of naphthalene [kg/m²·s] - Nu Nusselt number: Nu = d _h/ - P pressure of naphthalene vapor [Pa] - p non-dimensional pitch of in-line vortex generators: p = l _VG/S ₂ - Pr Prandtl number [–] - Q heat transfer rate [W] - R universal gas constant [m²/s²·K] - Re Reynolds number: Re = ·u _max·d _h/ - S ₁ transversal pitch between flat tubes [m] - S ₂ longitudinal pitch between flat tubes [m] - Sc Schmidt number [–] - Sh Sherwood number [–]: Sh = hd _h/D _naph - Sr staggered ratio [–]: Sr = (2Hsin – C)/(2Hsin) - T _p fin spacing [m] - T temperature [K] - u _max maximum velocity [m/s] - u average velocity of air [m/s] - V volume flow rate of air [m³/s] - x,y,z coordinates [m] - z sublimation depth[m] - heat transfer coefficient [W/m²·K] - heat conductivity [W/m·K] - viscosity [kg/m²·s] - density [kg/m³] - attack angle of vortex generator [°] - time interval for naphthalene sublimation [s] - fin thickness, distance between two VGs around the tube [m] - small interval - C distance between the stream direction centerlines of VGs - p pressure drop [Pa] - 0 without VG enhancement - 1, 2, I, II fin surface I, fin surface II, respectively - atm atmosphere - f fluid - fin fin - local local value - m average - naph naphthalene - n,b naphthalene at bulk flow - n,w naphthalene at wall - VG with VG enhancement - w wall or fin surface     

Thermal characteristics of the wake shear layers from a slightly heated circular cylinder

In this paper, we investigate the thermal characteristics of wake shear layers generated by a slightly heated circular cylinder. Measurements of the fluctuating temperature were made in the region x/d = 0.6 to x/d = 3 (where x is the downstream distance from the cylinder axis and d is the cylinder diameter) using a single cold-wire probe. The Reynolds number Re was varied in the range 2,600–8,600. For Re = 5,500, simultaneous measurements were made with a rake of 16 cold wires, aligned in the direction of the mean shear, at x/d = 2 and 3. The results indicate that the passive temperature can be an effective marker of various instabilities of the wake shear layers, including the Kelvin–Helmholtz (KH) instability. The temperature data have confirmed the approximate Re ^m dependence of the KH instability frequency (f _KH) with different values of m over different ranges of Re, as reported previously in the literature. However, it is found that this power-law dependence is not exact, and a third-order polynomial dependence appears to fit the data well over the full range of Re. Importantly, it is found that the wake shear-layer instabilities can be grouped into three categories: (1) one with frequencies much smaller than the Bénard–Kármán-vortex shedding frequency, (2) one associated with the vortex shedding and (3) one related to the KH instability. The low-frequency shear-layer instabilities from both sides of the cylinder are in-phase, in contrast to the anti-phase high-frequency KH instabilities. Finally, the observed streamwise decrease in the mean KH frequency provides strong support for the occurrence of vortex pairing in wake shear layers from a circular cylinder, thus implying that both the wake shear layer and a mixing layer develop in similar fashion.     

Experimental investigations of the stability limit of the helical flow of pseudoplastic liquids

The stability of the laminar helical flow of pseudoplastic liquids has been investigated with an indirect method consisting in the measurement of the rate of mass transfer at the surface of the inner rotating cylinder. The experiments have been carried out for different values of the geometric parameter = R ₁/R ₂ (the radius ratio) in the range of small values of the Reynolds number,Re < 200. Water solutions of CMC and MC have been used as pseudoplastic liquids obeying the power law model. The results have been correlated with the Taylor and Reynolds numbers defined with the aid of the mean viscosity value. The stability limit of the Couette flow is described by a functional dependence of the modified critical Taylor number (including geometric factor) on the flow indexn. This dependence, general for pseudoplastic liquids obeying the power law model, is close to the previous theoretical predictions and displays destabilizing influence of pseudoplasticity on the rotational motion. Beyond the initial range of the Reynolds numbers values (Re>20), the stability of the helical flow is not affected considerably by the pseudoplastic properties of liquids. In the range of the monotonic stabilization of the helical flow the stability limit is described by a general dependence of the modified Taylor number on the Reynolds number. The dependence is general for pseudoplastic as well as Newtonian liquids.Nomenclature C _i concentration of reaction ions, kmol/m³ - d = R ₂ –R ₁ gap width, m - F _M () Meksyn's geometric factor (Eq. (1)) - F ₀ Faraday constant, C/kmol - i _l density of limit current, A/m³ - k _c mass transfer coefficient, m/s - n flow index - R ₁,R ₂ inner, outer radius of the gap, m - Re = V _m ·2d·/µ _m Reynolds number - Ta _c = _c ·d^3/2·R ₁ ^1/2 ·/µ _m Taylor number - Z _i number of electrons involved in electrochemical reaction - = R ₁/R ₂ radius ratio - µ apparent viscosity (local), Ns/m² - µ _m mean apparent viscosity value (Eq. (3)), Ns/m² - µ _i apparent viscosity value at a surface of the inner cylinder, Ns/m² - density, kg/m³ - _c angular velocity of the inner cylinder (critical value), 1/s     

Precision of Hole-Drilling Residual Stress Depth Profile Measurements and an Updated Uncertainty Estimator

Background

Measurement precision and uncertainty estimation are important factors for all residual stress measurement techniques. The values of these quantities can help to determine whether a particular measurement technique would be viable option.

Objective

This paper determines the precision of hole-drilling residual stress measurement using repeatability studies and develops an updated uncertainty estimator.

Methods

Two repeatability studies were performed on test specimens extracted from aluminum and titanium shot peened plates. Each repeatability study included 12 hole-drilling measurements performed using a bespoke automated milling machine. Repeatability standard deviations were determined for each population. The repeatability studies were replicated using a commercially available manual hole-drilling milling machine (RS-200, Micro-Measurements). An updated uncertainty estimator was developed and was assessed using an acceptance criterion. The acceptance criterion compared an expected percentage of points (68%) to the fraction of points in the stress versus depth profile where the measured stresses ± its total uncertainty contained the mean stress of the repeatability studies.

Results

Both repeatability studies showed larger repeatability standard deviations at the surface that decay quickly (over about 0.3 mm). The repeatability standard deviation was significantly smaller in the aluminum plate (max ≈ 15 MPa, RMS?≈?6.4 MPa) than in the titanium plate (max ≈ 60 MPa, RMS?≈?21.0 MPa). The repeatability standard deviations were significantly larger when using the manual milling machine in the aluminum plate (RMS?≈?21.7 MPa), and for the titanium plate (RMS?≈?18.9 MPa).

Conclusions

The single measurement uncertainty estimate met a defined acceptance criterion based on the confidence interval of the uncertainty estimate.

     

Experimental determination of the free-stream disturbance field in a short-duration supersonic wind tunnel

The free-stream disturbance field in a short-duration supersonic wind tunnel is investigated at a nominal Mach number of Ma=2.54. A specially designed constant-temperature anemometer is used to be able to draw a complete fluctuation diagram within one wind tunnel run (testing time: 120 ms). It is shown that the disturbance field is dominated by acoustic waves radiated from the turbulent boundary layer on the nozzle and the sidewalls, like in conventional supersonic wind tunnels. The acoustic field appears to be composed of highly localized shivering Mach waves superimposed on a background of eddy Mach waves.Abbreviations  a constant in the thermal conductivity/temperature power law of air: k/k_r=(T/T_r)^a (dimensionless) - b constant in the viscosity/temperature power law of air: /_r=(T/T_r)^b (dimensionless) - Be bandwidth (Hz) - A, B constants in the wire heat transfer relation (Eq. (7), dimensionless) -   (dimensionless) - c_p specific heat at constant temperature (kJ/kg K) - c_v specific heat at constant volume (kJ/kg K) -  boundary layer thickness (m) - D function of the overheat ratio (dimensionless) - e anemometer output voltage (V) - _F end-loss attenuation factor for mass flow sensitivity (dimensionless) - _G end-loss attenuation factor for total temperature sensitivity (dimensionless) -  recovery factor (dimensionless) - f frequency (Hz) - f₁ normalized frequency (dimensionless) - F anemometer nondimensional sensitivity to mass flow fluctuations (dimensionless) - G anemometer nondimensional sensitivity to total temperature fluctuations (dimensionless) - F_AC F×F (dimensionless) - G_AC G×G (dimensionless) - f,g functions in the wire heat transfer relation (Eq. (7), dimensionless) -  c_p/c_v (dimensionless) - k thermal conductivity of air (W/m K) - k_r thermal conductivity of air at temperature T_r (W/m K) - k anemometer sensitivity to total temperature fluctuations (V/K) - l Mach rhombus half-length (Fig. 1, m) - Ma Mach number (dimensionless) -  viscosity of air (kg/m·s) - _r viscosity of air at temperature T_r (kg/m·s) - n constant in the wire heat transfer relation (Eq. (7), dimensionless) - Nu Nusselt number (dimensionless) - p pressure (Pa) - p₀ stagnation pressure (Pa) - r –F/G (dimensionless) - R unit Reynolds number (1/m) - Re Reynolds number (dimensionless) -  correlation coefficient between mass flow and total temperature fluctuations (dimensionless) -  density (kg/m³) - T time span (s) - T₀ total temperature (K) - T_r reference temperature (K) - T_w hot wire temperature (K) -  overheat ratio: =(T_w–T₀)/T₀ (dimensionless) -  –<e>/G (%) - u x-component of the flow velocity (m/s) - u_s source velocity at acoustic origin (m/s) - u inviscid velocity at acoustic origin (m/s) - x wind tunnel axis (Fig. 1, m)Symbols  x̄ temporal mean value of a fluctuating quantity x - x fluctuating part of x: x=x–x̄ - x_RMS^' root mean square of x - <x> x_RMS^'/x̄ - (X) relative uncertainty of a random variable X

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Sensitivity Study of Simulation Parameters Controlling CO<Subscript>2</Subscript> Trapping Mechanisms in Saline Formations

The primary purpose of this study is to understand quantitative characteristics of mobile, residual, and dissolved CO₂ trapping mechanisms within ranges of systematic variations in different geologic and hydrologic parameters. For this purpose, we conducted an extensive suite of numerical simulations to evaluate the sensitivities included in these parameters. We generated two-dimensional numerical models representing subsurface porous media with various permutations of vertical and horizontal permeability (k _v and k _h), porosity (f{\phi}), maximum residual CO₂ saturation (S_gr^max{S_{\rm gr}^{\max}}), and brine density (ρ _br). Simulation results indicate that residual CO₂ trapping increases proportionally to k_v, k_h, S_gr^max{k_{\rm v}, k_{\rm h}, S_{\rm gr}^{\max}} and ρ _br but is inversely proportional to f.{\phi.} In addition, the amount of dissolution-trapped CO₂ increases with k _v and k _h, but does not vary with f{\phi } , and decreases with S_gr^max{S_{\rm gr}^{\max}} and ρ _br. Additionally, the distance of buoyancy-driven CO₂ migration increases proportionally to k _v and ρ _br only and is inversely proportional to k_h, f{k_{\rm h}, \phi } , and S_gr^max{S_{\rm gr}^{\max}} . These complex behaviors occur because the chosen sensitivity parameters perturb the distances of vertical and horizontal CO₂ plume migration, pore volume size, and fraction of trapped CO₂ in both pores and formation fluids. Finally, in an effort to characterize complex relationships among residual CO₂ trapping and buoyancy-driven CO₂ migration, we quantified three characteristic zones. Zone I, expressing the variations of S_gr^max{S_{\rm gr}^{\max}} and k _h, represents the optimized conditions for geologic CO₂ sequestration. Zone II, showing the variation of f{\phi} , would be preferred for secure CO₂ sequestration since CO₂ has less potential to escape from the target formation. In zone III, both residual CO₂ trapping and buoyancy-driven migration distance increase with k _v and ρ _br.     

Comparison of the Calculated Results for Hypersonic Flow Past Blunt Bodies with the OREX Flight Test Data

The physico-chemical hypersonic air flow model is verified for the conditions of the experimental vehicle OREX reentry path over the altitude range H = 84 –105 km. The calculations are performed on the basis of both the viscous shock layer equations and the complete system of Navier-Stokes equations. The temperature and chemical component concentration fields calculated within the framework of the two models are in good agreement for H 100 km (Re 300). The numerical results for the heat flux at the stagnation point and the electron concentration agree well with the flight test data over the entire altitude range considered.