Selected Papers from the WSEAS Conferences in Spain, September 2008 Santander, Cantabria, Spain, September 23-25, 2008

Transcription

1 Comparisons between Velocit Profiles According to the Modified and the Navier-Stokes Equations of Motion and the Eperimental Measurements for Laminar Boundar Laer Over a Flat Plate Matthew J. Inkman, and Siavash H. Sohrab Robert McCormick School of Engineering and Applied Science Department of Mechanical Engineering Northwestern Universit, Evanston, Illinois 6008 UNITED STATES OF AMERICA matthewinkman007@u.northwestern.edu Abstract:- The solution of the modified form of the equation of motion is compared with the classical Blasius solution, eact numerical solution of the Navier-Stokes equation, and the eperimental observations of Burgers and Zijnen (194), Hansen (1930), Nikuradse (194), Dhawan (1953), and Büttner and Czarske (005) for laminar boundar laer flow over a flat plate. Nearl all of the eperimental data, ecept some of the data of Nikuradse, obtained b three different velocit measurement techniques of hot wire anemometr, Pitot tubes, and laser Doppler interferometr are in close agreement with the modified theor and show small but sstematic deviation from the classical theor. Also, the predicted friction coefficient of the modified theor is found to be in closer agreement with the eperimental observations of Janour (1951) as compared to that of the Navier-Stokes equation of motion. Ke-Words: - Laminar boundar laer. Modified form of the equation of motion. Navier-Stokes equation. 1 Introduction A scale-invariant model of statistical mechanics was recentl introduced [4] and applied to derive the invariant forms of conservation equations [5-6]. As an eample of an eact solution of the modified form of the equation of motion, the classical problem of Blasius [7] for laminar boundar laer flow over a flat plate was investigated and the predicted velocit profile was compared with some of the eperimental data available in the literature [8]. In the present stud, a more comprehensive comparison between the predicted velocit profile of the modified form of the equation of motion and the eperimental data of Burgers and Zijnen [9-11], Hansen [1], Dhawan [13], Büttner and Czarske [14] and Nikuradse [15] will be presented. In addition, the predicted friction coefficient of the modified theor will be compared with the eperimental measurements of Janour [16]. Finall, the predicted velocit profile will be compared with the eact numerical solution of the Navier-Stokes equation of motion. It is shown that with the eception of some of Nikuradse s data, all eperimental data agree with the prediction of the modified theor but sstematicall deviate from that of the classical Navier-Stokes theor. Scale-Invariant Forms of the Conservation Equations for Reactive Fields Following the classical methods [1-3], the invariant definitions of the densit ρ, and the velocit of atom u, element v, and sstem w at the scale are given as [4] ρ = nm = m fdu, u = v 1 (1) v = ρ 1 m f d u u, w = v +1 () Also, the invariant definitions of the peculiar and the diffusion velocities have been introduced as [4] V = u v, V = v w = V + 1 (3) ISSN: ISBN:

2 The scale-invariant model of statistical mechanics for equilibrium fields of... edd-, cluster-, molecular-, atomic-dnamics... at the scale = e, c, m, a, and the corresponding non-equilibrium laminar flow fields are schematicall shown in Fig.1. (J + 1) EED (J) ECD (J - 1) EMD FLUID ELEMENT EDDY CLUSTERS L c EDDY CLUSTER MOLECULES L m CLUSTER MOLECULE PARTICLES L a HYDRODYNAMIC SYSTEM FLUID ELEMENT EDDY CLUSTER FLUID ELEMENT L e λ e EDDY L c λ c CLUSTER L m λ m MOLECULE L a λ a EDDIES w e = v h v e = u h u e = v c CLUSTERS w c = v e v c = u e u c = v m MOLECULES w m = v c v m = u c u m = v a ATOMS w a = v m v a = u m u a = v s (J + 3/) LED (J + 1/) LCD (J - 1/) LMD (J - 3/) LAD Fig.1 Hierarch of statistical fields for equilibrium edd-, cluster-, and molecular-dnamic scales and the associated laminar flow fields. Following the classical methods [1-3], for constant transport coefficients with Sc = Pr = 1, the scale-invariant forms of the conservation equations were introduced as [5, 6] ρ t ρ D ρ. = Ω + w. (4) T + w T T h /(ρ c ). α = % Ω (5) p t v + w v v t p. ν = ρ 1 v Ω + ν (.v ) (6) 3 ρ An important feature of the modified equation of motion (6) is that it is linear since it involves a convective velocit w that is different from the local fluid velocit v. The original form of the Navier- Stokes equation with constant transport coefficients is nonlinear [1, ] v v. v = p + ν v + ν (. v ) (7) t ρ 3 In (7) the viscous term arises as a force from the stress tensor and hence the right hand side of the equation of motion. In (6) on the other hand, the viscous term arises from the convective term in the left hand side of the equation of motion and is associated with diffusional flu of momentum [6]. 4. Modified Theor of Laminar Boundar Laer Over a Flat Plate In this section, the solution of the modified equation of motion (6) for the classical problem of laminar flow within the boundar laer adjacent to a flat plate is considered [8]. The convective velocit field (w, w ) outside of the boundar laer, schematicall shown in Fig., w = 1 w = 1 δ( ) 0 1 Fig. Laminar boundar laer over a flat plate. is known and given b w = wo w = 0 (8) The conventional boundar laer assumption / / is introduced along with the dimensionless velocities ( v, v, w, w ) = ( v, v, w, w )/ w (9) o and coordinates = e / lh, / lh =, l H = ν / w o (10) where l H is the hdrodnamic diffusion length. For an incompressible fluid, under negligible pressure ISSN: ISBN:

3 gradient and in the absence of chemical reactions Ω = 0, the stead forms of (4) and (6) b (8) reduce to [8] v v + = 0 v v w = that are subject to the boundar conditions 0 = (11) (1) v = v = 0 (13) v = w = 1 (14) Because usuall l H = ν / w o 1, the boundar laer coordinates (, ) in (10) are stretched coordinates. The presence of boundar laer results in the transverse displacement of the outer flow field awa from the plate. According to (13)-(14), the local velocit v within the boundar laer must vanish at the plate and match the outer convective velocit field w = 1 at the edge of the boundar laer, i.e. in the limit (Fig.). Therefore, the convective velocit within the boundar laer will be taken to have the constant value of w = ½ at all aial locations. Introducing the value w = ½ into (1)-(14) leads to the solution [8] v = erf ( ξ ) (15) that involves the similarit variable ξ = (16) To facilitate the comparisons, the solution (15) is epressed as v = erf[ η / ], η = / (17) in terms of the same similarit variable η as in the classical theor [, 7]. The boundar laer thickness is obtained from (15) as δ = (18) 1/ Re that is in close agreement with the classical numerical result of Blasius [, 7] δ 5.0 Re (19) One can epress the solution (15) in terms of the stream function [8] (0) 0 ξ Ψ = erf ( ξ)dξ The transverse velocit that satisfies (11) is obtained from (0) as v = ξerf ( ξ) erf (z)dz ξ (1) 0 The magnitude of the transverse velocit at the edge of the boundar laer is obtained from (1) as 1/ 1/ v ( ) = 0.798Re () π that is in close agreement with the classical result of Blasius [7] v ( ) = 0.861Re (3) 1/ where the Renolds number is Re = = w o / ν. The plot of the transverse velocit (1) shown in Fig.3 Re v Fig.3 Transverse velocit component calculated from (1) for the boundar laer over a flat plate. has the asmptotic value of / π = instead of of the classical result given in Fig.7.8 of Schlichting []. 5. Comparisons between the Theor and the Eperimental Observations The predicted velocit profile (15) is first compared with the eperimental data of Burgers and Zijnen [9-11] obtained in the Aeronautical Institute of the ISSN: ISBN:

4 Technical Universit Delft. In these investigations, the boundar laer velocit profiles were obtained for the flow of air over a flat plate at multiple free-stream velocities and locations from the leading edge of the plate using hot-wire anemometr. The measurements were conducted in a 0.8m 0.8m wind tunnel at in the Laborator for Aero- and Hdro-Dnamics at Delft, emploing a cm diameter copper anemometer wire. The velocit profiles shown in Figs.4a-c Fig.4 Comparison between the eperimental data of Burgers and Zijnen [9-11] and the predictions of Blasius and modified theories. were obtained for the free stream velocities w o = 1., 1.6,.4 cm/s at particular distances = 5, 10, 0 cm from the leading edge of the plate. As can be seen in Figs. 4a-4c, the measured velocit profiles displa the characteristic shape of the modified profile, while differing noticeabl from the predicted profile of Blasius. It is emphasized that the deviations of data in Fig.4 from classical theor are sstematic as stated b Burgers [9] The eperimental curves generall lie above Blasius s curve. For ever value of V the deviation increases with the section, which shows that we have to deal with a real phenomenon, as the increase of is accompanied b an increase of δ, so that the influence of the correction for the cooling effect of the wall will be diminished. Moreover, it occurs in the same wa with all values of V. (a) (b) (c) It should be noted that further along the plate, the boundar laer profile slightl eceeds the modified curve s value; this is likel due to the development of boundar laers on the walls of the wind tunnel itself, resulting in the restriction of the total flow through a smaller cross-sectional area. This contributes to a slight increase in the free-stream velocit above its nominal value farther along the plate, which accounts for the higher profile seen at greater distances from the leading edge. Indeed, as was stated b Hansen [1], Zijnen himself established the increase in velocit along the plate, though for another purpose (Thesis, Report No.6, Delft, 194, pp.39-4). However, as was mentioned previousl, even these slightl increased velocit profiles noticeabl conform to the shape of the modified profile. We net eamined the eperimental findings of Hansen [1]. In his investigation, the boundar laer velocit profiles were obtained for flow of air over a flat plate with either sharp or thick edge. The measurements were conducted in a small wind tunnel belonging to the Aeronautical Institute of the Aachen Technical Universit with the diameter of entrance cone of 30 cm and the length of the free jet between the entrance and eit cones of.5 m [1]. As emphasized b Hansen [1] the eperiments were performed for a free stream where the static pressure remained constant as opposed to the confined stream in the stud of Burgers and Zijnen [9-11] discussed above. Also, instead of hot wire anemometr, a small Pitot tube with an outside diameter of 0.35 mm and ISSN: ISBN:

5 inside diameter of 0.31 mm was used for most of the measurements. The pressure was determined b means of two alcohol pressure gauges. The measurements could onl be made in the velocit range 4 to 37 m/s, since the tubes vibrated strongl at higher speeds. The measured velocit profile for plate with sharp leading edge with free stream velocit w o = 8 m/s is shown in Fig.5a. (a) The results of the measurements for plate with thick leading edge at various aial positions from the leading edge are shown in Fig.5b and Fig.5c respectivel corresponding to the free-stream velocities w o = 8 and 16 m/s. As stated b Hansen [1] the eperimental data showed deviations due to the leading edge effects and such effects were more pronounced for the plate with sharp leading edge. As shown in Figs.5b-5c for plates with thick leading edge the measured velocit profile obtained b this entirel different eperimental apparatus once again agree with the prediction of the modified theor while deviating from that of the classical theor. Also, for the plate with sharp edge in Fig.5a, if one removes the data at small values due to the leading edge disturbances once again one finds close agreement the predictions of the modified theor. Moreover, the deviations of the data of Hansen [1] from the classical theor in Fig.5 appear quite similar to those found independentl b Burgers and Zijnen [9-11] shown in Fig.4. Net the eperimental data of Dhawan [13] are compared with the predictions of the classical and modified theories as shown in Fig.6. (b) Fig.6 Comparisons between the eperimental data of Dhawan [13] and the predictions of Blasius and modified theories. (c) Fig.5 Comparisons between the eperimental data of Hansen [1] and the predictions of Blasius and modified theories. Once again, the eperimental data of Dhawan in Fig.6 are in closer agreement with the modified theor and the deviate from the classical theor. Also, the deviations of Dhawan s data from the classical theor in Fig.6 are quite similar to those of the independent studies b Burgers and Zijnen [9-11] and Hanson [1] shown in Figs.4 and 5. ISSN: ISBN:

6 A fourth source of eperimental data is the recent paper of Büttner and Czarske [14]. In their eperiment, broad-area laser diodes were emploed in laser doppler interferometr to measure the boundar laer velocit profile of flow over a flat plate. The eperiment was conducted in a 0.1m 0.1m Eiffeltpe wind tunnel, and the flow was characterised b tracking diethlhealsebacathe tracer particles with a mean diameter of.5 µm. Their results are compared to the modified and Blasius profiles in Fig.7. eperimental data of Nikuradse [15] are found to alwas locate on the lower boundar of the more recent data of Dhawan shown in Fig.6. In the light of the observations of Burgers and Zijnen [9-11], Hansen [1], Dhawan [13], and Büttner and Czarske [14], it is clear that the data of Nikuradse shown in Fig.8 are biased towards the classical theor. In fact, as stated b Schlichting [], Nikuradse introduced certain corrections to account for what he believed were leading edge effects resulting in the total absence of scatter in his data shown in Fig.8. However, since the similarit solution of Blasius [7] is singular at = 0, due to the similarit variable (16), and hence onl valid awa from the leading edge of the plate, it is suspected that Nikuradse s corrections were not warranted. v Fig.7 Comparisons between the eperimental data of Büttner and Czarske [14] and the predictions of Blasius and modified theories. Error was introduced into the measurement due to variation in the doppler frequenc and the interference fringe spacing, especiall near the edge of the measurement volume. This seems to have resulted in significant down-side bias in the middle portion of the velocit profile, as is evident from the large number of low outliers in this region. However, while acknowledging this dip in the central portion of the data, the overall shape of the eperimental datapoints, Fig.7, conforms more closel to the modified profile than to that of Blasius, a correspondence that is nearl eact in first and final third of the dataset, outside of the area of the lowvelocit dip. Finall, the fifth set of eperimental data is the well-known results of Nikuradse s [15] eperiments and comparisons between his data and the predictions of Blasius [7] and modified [8] theories are shown in Fig.8. In accordance with Fig.7.9 of Schlichting [], the eperimental data of Nikuradse [15] ver closel follow the numerical calculations of Blasius [7] and hence the dashed line in Fig.6. Therefore, the h Fig.8 Comparisons between the eperimental data of Nikuradse [15] and the predictions of (1) modified theor () Blasius theor. It is possible that the close agreement between data and the classical theor both near the wall and near the edge of the boundar laer, see Figs.4-8, contributed to Nikuradse s decision to correct the data that somewhat deviated from the theor at the intermediate distances from the wall. However, the consistenc and similarit of deviations of the data from classical theor occurring in four independent eperiments, Figs.4-7, suggest that the cannot be solel attributed to the geometr of the leading edge. Having found the modified theor to more accuratel predict the longitudinal velocit profile of the boundar laer as measured b all three major boundar laer velocit measurement techniques, i.e. Pitot tubes, hot-wire anemometers, and laser Doppler ISSN: ISBN:

7 interferometr, our confidence in the accurac of the modified theor s predictions is increased. Net, the predicted friction coefficients of the modified theor [8] C = / π = Re (4) f 1/ 1/ and the classical theor of Blasius [, 7] C f = 0.664Re (5) 1/ are compared with the eperimental measurements of the variation of the friction coefficient with Renolds number obtained b Janour [16] as shown in Fig.9. The skin friction measurements were conducted in an oil tunnel of 1.79m 1.48m cross-section, using the deflection of a pendulum scale to characterize the shear force at the plate. Having seen the correlation of the modified theor to eperimental data, we net compare the predicted velocit profile with that obtained b numericall solving the full Navier-Stokes equations. A first attempt at obtaining computational results was made emploing the Fluent software package. A mesh of a flat plate was generated in Gambit, with a plate length of 0.5 m, and 0.1 m gap before the leading edge of the plate, and a height of 0.5 m. The plate mesh had an interval count of 100 with a ratio of 1.08 for both the horizontal and vertical meshing parameters, while the area before the leading edge of the plate had an interval count of 10 with a ratio of 1.1. The left edge of the mesh was specified as a velocit inlet. The right and top edges were defined as pressure outlets. The bottom edge before the plate was assigned a smmetr BC, while the bottom edge at the plate itself was simpl defined as a wall. The mesh was then imported into Fluent and solved for a flow of a viscous, incompressible fluid with the properties of water. The inlet velocit was specified to be 0.10 m/s, parallel to the plate. Solutions were found at 0.1 m, 0.3 m, and 0.5 m from the leading edge of the plate. These results are displaed in Fig.10. Fig.9 Friction coefficient versus Renolds number measured b Janour [16] compared with the predictions of Blasius and modified theories. As can be seen in Fig.9, the line of the modified theor runs directl through the measured data, while the Blasius line acts as a lower bound for the measurements. This suggests that the modified profile will more accuratel predict the skin friction coefficient over the full range of laminar flows. 6 Comparisons Between the Theor and Eact Numerical Solution of the Navier-Stokes Equation Fig.10 Comparison between the numerical solution of the Navier-Stokes equation using the Fluent and the predictions of Blasius and modified theories. As can be seen from Fig.10, the Fluent solutions fall eactl along the classical Blasius profile. Hence the Blasius profile is indeed an accurate approimate solution of the full Navier-Stokes equations. ISSN: ISBN:

8 To confirm the results in Fig.10, a second numerical solution of the N-S equations was sought. The additional numerical solution emploed the NPARC Alliance Flow Simulator. The simulator is applied to flow over a flat plate with parameters as emploed in the laminar flat plate stud in the validation section of the NPARC website [17]. A mesh with stream wise grid spacing of 0.05 and a normal spacing of 0.4 was used in terms of η coordinate. Initial conditions were set for a flow of Mach number 0.1 with static pressure 6.0 psia and temperature 900 R. A viscous wall boundar condition is applied at the plate, while an inviscid wall boundar condition is applied to the region before the plate, from i = 1 to 10. The left edge has an inflow/outflow BC, and the right edge is a confined outflow. The far field boundar at top is specified inflow/outflow. The results of the simulation as presented in the validation section of the website are shown in Fig.11. not completel represent the flow within the boundar laer. The predicted profile derived from the modified theor could potentiall be improved in accurac b an iterative calculation. The current form of the velocit profile v was derived based on a linear velocit profile within the boundar laer, hence w = ½. This first approimation of the profile could be improved b averaging it from the plate to the edge of the boundar laer from (3), making this new average w, and solving again for the profile. This process could be iterated until the solutions converge. 7 Concluding Remarks While it ma initiall seem to be a dramatic statement, the idea that the classical Navier-Stokes equation ma be supplanted b the new invariant momentum conservation equation appears to be supported b the eperimental data. The Blasius profile is sufficientl close to eperimental observations that in the past, an discrepancies were generall attributed to eperimental error. However, we have seen in this stud that the deviation from the Blasius profile is a persistent problem across all major eperimental methods of boundar laer velocit measurement and eists for skin-friction measurements, as well. Given the conceptual advantages of the modified theor over Navier- Stokes, and the fact that it far more accuratel reflects actual eperimental results, it ma indeed be time to reevaluate the fundamental equations of fluid mechanics. References : Fig.11 Comparison between the eact solution of Navier-Stokes and the Blasius solution. As can be seen in Fig.11, the solution falls eactl on the Blasius profile, once again confirming that the Blasius solution is an accurate solution of the full Navier-Stokes equations. This is a point of some interest. As we observed above, the modified theor s solution more accuratel reflects actual eperimental results, while the Blasius solution accuratel reflects the numerical solutions of the Navier-Stokes equations. The natural conclusion from this is that the invariant form of the momentum conservation equation (6) is actuall the proper form, while the classical Navier-Stokes equation (7), does [1] de Groot, R. S., and Mazur, P., Nonequilibrium Thermodnamics, North-Holland, 196. [] Schlichting, H., Boundar-Laer Theor, McGraw Hill, New York, [3] Williams, F. A., Combustion Theor, nd Ed., Addison-Wesle, New York, [4] Sohrab, S. H., A scale-invariant model of statistical mechanics and modified forms of the first and the second laws of thermodnamics. Rev. Gén. Therm. 38, (1999). [5] Sohrab, S. H., WSEAS Transactions on Mathemathics, Issue 4, Vol.3, 755 (004). [6] Sohrab, S. H., Derivation of Invariant forms of Conservation Equations from the Invariant Boltzmann Equation. in Theoretical and Eperimental Aspects of Fluid Mechanics, S. H. ISSN: ISBN:

9 Sohrab, H. C. Catrakis, and F. K. Benra (Eds.), WSEAS Press, 7-35, 008. [7] Blasius, H., Grenzschichten in Flüssigkeiten mit kleiner Reibung. Z. Math. Phs. 56, 1 (1908). English translation in NACA TM 156 (1950). [8] Sohrab, S. H., Modified theor of laminar boundar laer flow over a flat plate. IASME Transactions, No.8, 1389 (005). [9] Burgers, J. M., The Motion of a Fluid in the Boundar Laer along a Plane Smooth Surface, Proceedings of the First International Congress for Applied Mechanics, Delft, (194). [10] Burgers, J. M., and van der Hegge Znen, B. G., Preliminar measurements of the distribution of the velocit of a fluid in the immediate neighborhood of a plane smoooth surface, Verhand. Akad. v. Wetensch. Amsreadam (ie sectie) Deel XIII, No.3, 194. [11] van der Higge Zijnen, B. G., Measurements of the velocit distribution in the boundar laer along a plane surface, Thesis Delft, 194. [1] Hansen, M., Velocit distribution in the boundar laer of a submerged plate. NACA TM, N o 585, [13] Dhawan, S., Direct measurements of skin friction, NACA TN 567 (1953) [14] Büttner, L. and Czarske, J., Investigation of the influence of spatial coherence of a broad-area laser diode on the interference fringe sstem of a Mach Zehnder interferometer for highl spatiall resolved velocit measurements, Applied Optics, 44, 9 (005). [15] Nikuradse, J., Laminare Reibungsschichten an der längsangetrömten Platte. Monograph, Zentrale f. wiss. Berichtswesen, Berlin, 194. [16] Janour, Z., Resistance of a plate in parallel flow at low Renolds numbers, NACA TM 1316, (1951). [17] Laminar Flat Plate, NPARC Alliance Validation Archive, NPARC, Feb. 0, am/fplam.html ISSN: ISBN: