Turbulence Models and Their Application in Hydraulics 2nd

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Turbulence Models and Their Application in Hydraulics：A State-of-the-Art Review(2nd).
II I FOREWORD by Chao-Lin Chiu, Prof essor of Civil Engineering University of Pittsburgh, Pittsburgh, Pennsylvania, U.S.A. Mr. J.E. Prins, Secretary of lAHR, invited me to write this Foreword as this book is presented by the Sect ion on Fundamentals of Division lI and I happen to be the Chaiman of the Division. Under the leadership of Messrs. M. Hug(past President)and H..Schoemaker (former Secretary), the IAHR begon promoting publ ications of books by members. This book represents one of the first to be published as a part of the plan fo provide a better service to IAHR members. The book was reviewed, approved and edited by IAHR prior to publication The topic of this book is timely. The scope is broad that makes it difficult ho satis- Fy readers who may demand information in great depth or wide areas of practical applica tions in hydraulics. However, efforts were clearly made to present promising or proven turbulence models useful in calculating turbulence terms that appear in time-averaged equations governing moan-flow quantities, and to give examples to illustrate possible appli- chers in the field of hydraulics where applications of turbulence models are gaining momentum I am pleased that the Sections in Division II are very active in support of the pre sent (AHR effort to serve its members. There will be more publications of this nature coming out of our Division. I am fully aware of the fact that it is not an easy task to find a person who has both the qualifications and will ingness to work on a state-of-the-art review such a this. Efforts which Dr. Rodi put into this book should, therefore, deserve a propor e?precia www.docin.com CONTENTS EXAMPLES OF APPLICATIONS TO HYDRAULICS PROBLEMS PREFACE 3. 1 Channel Flow al Two-Dimens ional Calculations SUMMARY/RESUME i b Three dimensional effects 50 NOMENCLATURE 3.2 Recirculating Flow Due to Abrupt Changes in Channel eometry NTR○DUT。N 3, 3 Submerged Jets and Wakes 1.1 The Role of Turbulence Models a) Non-Buoyant Jets and Wakes b) Buoyant Jets and Wakes 1. 2 Scope and Outline of the Revi 3.4 Discharges into Nearly Stagnant Water TURBULENCE MODELLING a) Surface Discharges b) Submerged Discharges into Shallow Water 2.1 Mean -Flow Equations and the Problem of Closure 5 3.5 Discharges into Moving Streams 2.2 The Nature of Turbulence a Two=Dimensional Calculations 66 2. 3 Basic Concepts and Classification of Turbulence Models 10 b) Three-Dimensional Calculations 2. 4 Zero-Equation Models 14 3.6 Unsteady Flow Calculations o Constant Eddy Viscosity /DIffusivity 14 4 CONCLUSJONS b) Mixing Length Models 16 c Prandtl's Free Shear Layer Model 2.5 One-Equation Models APPENDIX A: Introduction to tensor notation a] Models Using the Eddy Viscosity Concept 20 APPENDIX B: The Length Scale Specification Proposed by b Bradshaw et als Model 24 c Heat and Mass Transfer Calculation and buoyancy Effecis APPENDIX C: Table 5: Summary of Turbulence Models and A Initial Conditions Used in the Application 2.6 Two-Equation Models 0 Length-Scale Equation W2 ocn REFERENCES 分)Thek- E Mod c) Nonisotropic Eddy Viscosity 30 d) Heat/Mass Transfer and Bu e) k-E Model for Depth-Average Calculations 3 f Asse 33 2.7 Turbulent Stress/Flux-Equation Models a) Reynolds-Stress Equations b) Equations for Scalar Fluxes ui 4p and Fluctuations p2 38 c) Algebraic Stress /Flu 4 d) Assessment 2.8 boundary Conditions 44 PREFACE of-the-art os/ w predictions. The so-called k-e turbulence model receives most attention in this and is recommended as the most suitable model af the present state Turbulent motions contribute significantly to the transport of momentum, heat and of-the- art; as I was personally involved in the development and application of this par mass in most flows of practical interest and therefare have a determining influence on the lar model, this may seem as a lack of objectiveness. I should like to state however thor icu distributions of velocity temperature and species concentrat ion over the flow field, It is the best of my knowledge there is no other model that has been tested successfully for the basic task of engineers working in the field of fluid mechanics to detemine these dis ly as many different hydraulic flow problems as has the k-E model tributions for a certain problem, and if the task is to be solved by a calculation method there is no way around mak ing assumptions about the turbulent transport processes. Basically The suggestion to write this review for the Internat ion al Association for Hydraulic this is what turbulence modelling is about because the turbulent transport processes cannot Research came from Dr, G. Abraham in his capacity as Chaiman of the Association's Com be calculated with an exaet method, they must by approx imoted by a turbulence model which mittee on Fundamentals, and i should like to thank him for this stimulus as well as for his with the aid of empirical information, allows the turbulent transport quantities to be relat valuable comments on earlier drafts. i should also like to thank professor H. Kobus ,secre ed to the mean升 low field tary of the Committee on Fundamentals, and Professor E. Naudescher, Chairman of the Division ll of IAHR and Director of my home Institute, for their constant support and advice Until recenTly, the available turbulence models did not appear very promising for My thanks are due further to Dr. L. Fink, dr M, F. Gauthier, Professor B.E. Launder, olving but the simplest real-life flow problems, and they were used particularly seldom Dr. J. McGuirk, Dr. A.K. Rastogi, Dr, U. Svensson, and Dr. D S. Trent for making the field of hydraulics. There, calculations were carried out mainly for so-called far-field available unpublished infomation and results. The effort of Dr. I.A. Sherenkov, who pro- situations where the turbulent transport is either relativety unimportant or can be approxi vided a short summary on turbulence-model work in the Soviet Union, is also great ly acknow mated to sufficient accuracy by crude assumptions. For near-field situations where turbulent ledged. Further, I am grateful to Dr. J. Mc Guirk and Dr C B, Vreugdenhil for helpful ce dominates the flow beha viour and requires refined modelling, hydraulic engineer comments on an earlier draft, to Dr.M. A, Leschziner for his careful proof reading, and sorted so far mainly to physical modelling. Over the last ten years however, advances in to Miss G. Bartman and Mrs. S, Issel for their great effort and patience in producing the computer technology have made it possible to thoroughly test increas ingly complex turbul manuscript. Finally I should like to acknowledge the financial support of the Deutsche Fe ence hypotheses and to apply them in practical calculations. As a result, the situat ion has schungsgemeinschaft. A Genman version of this review was accepted by the Fakultet Fur changed, and the theoretical treatment of near-field problems is now also within reach Bauingenieur-und Vemessungswesen of the University of Karlsruhe as Habilitation thesis although the advanced turbulence models presently available require further testing, it can already be stated that they will in many cases provide good estimates for problems that defied a theoretical treatment so far. To demonstrate this is one of the purposes of this book Most of the turbulence-model-development and application work was carried out Karlsruhe, March 1979 W. Rodi bulent transport processes ts equally important in many hydraulic flow problems. It is there- fore perhaps not easy for hydraulic engineers to acquire up-to-date knowledge about the existing turbulence models described mainly in the literature of other areas, and to judge the appl icability of these models to hydraulic flow problems The present book thereforc cin. com aims to provide an introduction to the subject of turbulence modelling in a form easy te understand for anybody with o basic background in fluid mechanics, and to summarize the present state of the art. the review deals specifically with certain modelling aspects pe- culiar ta hydraulic flows, but it is kept general enough to be of interest also to workers in other areas in which turbulent transport processes play an important role.I help hydraulic engineers judge the suitability of the various turbulence models for their problems, a review is given also of published model applications to hydraulic flow problems with the emphasis on near-field issues the more advanced also more pror turbulence models to draulic flow problems did not commence 1970 and was even then pursued by a few researchers only. Therefore the range of flows for which these models were tested so far is unfortunately somewhat restricted, and final judgement on the general applicability must await further testing. Also, most of the calculations presented for any one flow example were obtained with only one model. Comparisons of the perfomance of various models when applied to the same problem would certainly have been more valuable, but such compari sons have seldom been reported in the literature, and it was not wlthin the seope of this SUMMARY justifiable pour la plupart des problemes hydrauliques. L'emploi de tous ces modeles n'est recommand que pour A survey is given of existing mathematical models for describing the turbulent te. L'emploi d' un mode roblemes en"champ proche", n'interessont qu'un domaine limi transport of momentum, heat and mass in flows. The merits and demerits of the individual au sein de grandes masses d'eau ne serait guere justifie, puisque ces problemes s'abordent models are discussed with respect fo their predictive ability and computationa! economy bien mieux au moyen de coefficients d echange judicieusement choisis xamples of model applications to a fairly Targe variety of small-scale (nea-field hydrau- lics problems are presented, with emphasis on recently developed more refine he performance of the models is judged by comparison with experiments, and conclusions are drawn on t he range of appl icability of the individval models, The so-called two-equa- tian models employing differential transport equations for the veloc ity and length scales of the fluctuat ing motion are found to offer the best compromise between width applE cability and computational economy at the present state of develapment. In particular, the k-e model is shown to predict reasonably well a fairly large range of bas ic hydrau- lics problems with the same empirical input. Simpler models like the mixing length hypo- thesis or one-equations models using only a transport equat ion for the turbulence velocity scale are suitable mainly for shear-layer problems for which the length-scale distribution can be prescribed realistically. The more complex models employing transport equat ions for the individual turbulent stress es and heat or mass fluxes are conceptually superior, but they are at present insufficiently developed and tested, and the inereased computational effort they require seems hot justified for most hydraulics problems. All these models are recommended for application to small-scale problems only. Refine ranted for calculating the horizontal turbulent fransport in large woter bodies where the\ delling is not w use of suitably chosen exchange coefficients is more appropriate RESUME Passage en revue des diff erents modeles mathematiques destines d l'etude des phe omenes de transport turbulent des quantites de mouvem ent, therm iques ou massiques dans les fluides en acoulcment, Examen des advantages et des inconvenients de ges modeles, dans optique de leurs capacites isionell es et du volume des calculs necessaires. Pre mentation d'examples d appl ication des model es d l'etude de problemes hydrauliques rela- tivement diversifies et intervenant dans un champ limite(" champ proche"), notamment en ce qui concerne certains model es"affines"de conception recente. Appreciation des possibi- in. com lites des differents modeles, d la lumiere de comparaisons par rapport aux procedes experi mentaux, et conclusions relatives aux domaines utilisation de ces modeles. Les modeles dits"a deux equations", dont les echelles de vitesse et de longueur des phenomenes fluc tuants sont definies por des equations de transport differantielles, permetiraient &I'&tot actuel de leur developpement, d associer a mieux un domaine d'appllcation etendu des modeles a un volume minimal des calculs necessaires. En particulier, le modele k-E permet, a partir des memes donnees entree empiriques, une etude assez precise des problemes hydrauliques fondamontaux interessant des domaines relat iv ement etendus, les modeles plus simples, tels que repasant sur I'hypothese de lo longueur de melange, ou d une seule equation (une seule equation de transport, definissant I"echelle de vitesse de la turbulence)se pretent, essentiellement, a l'etude des probl emes relevant des couches d ecoulement en cisaillement, pour lesquels il est poss ible de se fixer une repartition rea- liste de Echelle des longueurs. Les modeles plus complexes, repas ant sur la definition des contraintes de turbulence et des flux thermiques ou mossiques par des equations de transport, sont superieur sur le plan conceptuel, mais ne sont, ni tout d fait au point, ni ffisamment eprouves; d'autre part artr ils necessitent des calculs d'une ampler difficile ent NOMENCLAT URE flux Richardson number R gradient Richardson nutmber, Lq.(2. 20) empirical constant in diffus ivity law(2.22 salinity B channel width radial coardinate discharge width / depth empirical constant in diffusivity law(2. 21) volumetric source/sink tem of quantity p concentration, temperature constans in turbulence models time D, d diameter or other geometrical parameter U,V,W mean velocity components in x, y, z direction entrainment rate friction parameter in Io (285) fluctuating velocity is in x, y, z directi distance between shear zones(Eq. B5) instantaneous or mean velocity component in x! direction densimetric Froude number u fluctuating velocity component in xi direcrion duction/ destruction of k U午,U Vtw/p friction velocity G buoyancy production/ destruction of u U UE free- stream veloc计t gravitational acceleration, eloc ity scale e furpulen gradient Xyz width of recirculation zone cc-ordingtes in tensor notation water depth half-width of jet or wake, def ined by cross-stream distance from the h water depth axis to a point where (U-Uoo=1/2(Umax -Uoo) J depth-average flux of p in direction x Ury/v=dimensionless wall distance k 1/2uiu= turbulent kir length scale of turbulence Prandtl mixing length WWW.docn己 empirical constant in (2.19) n Manning roughness Factor volumentric expansion coefficient, empirical constant in (2.19) stress production of k eddy diffu sivity production ofφz constant in pressure-strain model (2. 64) stress production shear-layer thickness fluctuating pressure Kronecker delta, 1 for i=i and=0 for ifi flow rate disslpation rate of k hea↑ flux at surface issipation rate of radius surface elevation elocity ratio ratio of time scales of scalar and velocity fluctuations e momentum thickness of jets Reynolds number 飞 von aman constant in log law(2.85 入 m。lecU usivity of scalar quantity 1。| NTRODUCTION pecifying the mixing length in boundary layers kinematic molecular viscosity 1.1 The Role of Turbulence models eddy (or turbulent )viscosit In hydraulics, as in other areas of fluid mechanics, the flows of practical rele pressure strain term vance are almost always turbulent; this maans that the fluid motion is highly random,un- steady and three-dimensional. Due to these complexities, the turbulent motion and the pressure-scalar-gradient correlation eat and mass-transfer phenomena associated with it are extremely difficult to describe and fluid densit thus to predict theoretically. Yet, the bas ic task of hydraulic engineering is that of pre- dicting water-flow phenomena, and, because predictions"by way of experiments are turbulent Prandtl / schmidt number usually very expensive, calculation methods are in great practical demand k, E constants in k-E In spite of all the recent advances in computer technology turbulent flows cannot s『e5 at present be cal culated with an exact method. The exact equations describing the tur- wall shear stress bulent motion are known (the Navier-Stokes equations), and numerical procedures are available to solve these equations but the storage capacity ond speed of present-day stress acting in direction xi on a face perpendicular to computers is still not sufficient to allow a solut ion for any practically relevant turbulent instantaneous or mean scalar quantity Flow. The reason is that the turbulent motion contains elements which are much smaller than the extent of the flow domain, typically of the order of 10-3 times smaller. To re- fluctuating scalar quantity solve the motion of these elements in a numerical procedure, the mesh size of the nu- merical grid would have to be even smaller; therefore at least 109 grid id points would Subscripts necessary to cover the flow domain in three dimensions, Storing the flow variables at so many grid points is still far beyond the capacity of the fast-access memory of present- b. b day computers, and, in addition, the number of arithmetic operations which would be re- quired is so large that the computing time would also be prohibitive. Saffman [T recent- discharge ly claimed that in about 20 years'time computers will have been developed with suffici external ent capacity to solve numerically the exact equations for turbulent flows; for the near future however an exact treatment of turbulence is certainly out of th river reference Until very recently there was not even the gl impse of a hope of ever solving the surface af ex计t case without buoyancy www.doci exact equations for turbulent flows, Since engineers nevertheless needed calculation methods, they took recourse to empirical and semi-empirical methods. Empirical methods simply correlate exper imental results and can therefore be used with confidence only for dircct interpolation of these results; the Chezy friction law is a typical example condition in external (free)stream With the aid of dimensional analysis, experimental dat a were condensed into many useful empirical fomulae. However, these can describe only the simplest phenamena of interest at centre-line and are not suitable for camplex geometries; for whenever there are more than just a few parameters detemining the problem in question, generally valid empirical correlations are difficult if not impossible to attain. Therefore, at an early stage, another approach was also followed and methods were developed which make use of our theoret ical know ledge about fluid-flow phenomena. These methods are based on the conservation laws for mass, momentum and energy and are therefore, at least potentially of greater ge- neral validity than strictly empirical relations. The basic conservation laws are expres- ed by the exact equat ions mentioned above, which describe all details of the fluid mo- tion. Because there was (and still is)little hope of solving these equations, and because engineers are in any case not interested in the details of the fluctuating motion, a statis- tical approach was taken(as first suggested by Osborne Reynolds)and the equat ions were averaged over a time scale which is long compared with that of the turbulent motion The resulting equations describe the distribution of mean velocity, pressure, temperature and species conc entration in the flow and thus the quantities of prime interest to the en- ineer. Unfortunately, the process of averaging has created a new problem now the 2 quations no longer constitute a closed system since contain unknown terms repre- mentum conservation can be reduced to ordinary ones. The partial differential equations senting the transport of mean momentum, heat and mass by the turbulent motion. The sy governing the more general flow situations could stem can be closed only with the aid of empirical input, whence the calculation methods few flows to which they could be applied, the early models lacked universality in that they required different empirical constants for different flows; the desire to remedy this led to the development of more complex models starting in the 1940s. these models Empirical information con be put into the sysrem of equations in two distinctly dIf ferent ways. Integral methods, which are suitable mainky for thin shear layers (boundary. give up the direct, algebraic link between the turbulent transport terms and tha mean- Flow quantities and emplay differential transport equations for turbulence quantities such layer-type flows), introduce empirical profile shapes so that the originally partial differen- as the kinetic energy of the turbulent motion. Some of these models are conceptually tidl equations can be reduced to ordinary ones. Further input is necessary which describ as advanced as the most complex turbulence models in use today. However, it took an- es the global effect of turbulence r like the entrainment laws for free shear flows and other 20 years before these modcls could actually be applied and tested. In the 1960's the friction ar energy dissipation laws for wall boundary layers. In contrast, the so-called computers became suffIclently powerful and, shortly thereafter, numerical technique field methods, which employ the original partial differentiol equations, require specifi- sufficiently well-developed to allow the partial differential equations for the mean flow cation of the turbulent transport fems appearing in the equations at each point in the Flow (and in complex turbulence models) also for turbulence quantities to be solved for man This specification is accomplished by a mathematical model of the furbulent transport pro flow situations, As a consequence, the main effort in developing, and almost all the cesses which is called a turbulence model". Therefore a turbulence model is defined as effort in testing turbulence models has been confined to the last 10 years a set of equations (algebraic or differentiol] which detemine the furbulent fransport terms in the mean-flow equations and thus close the system of equat ions. Turbulence models This development and the resuiting application of turbulence models was mainly are based on hypotheses about turbulent processes and require empirical input in the restricted to the area of mechanical and aeronautical engineering. various reviews are form of constants or functions; they do not simulate the details of the turbulent motion available on this work [2, 3, 4], which give a fairly comprehensive picture of the state- but only the effect of turbulence on the mean flow behaviour of-the-art in these fields, In hydraulic engineering however, the application of turbul- ence models was, until very recently, restricted to the early most simple models Turbulent transport processes are strongly problem-dependent; for example, they depend on geometrical conditions of large and small scale(e. g. well shape and rough- As advanced turbulence models have shown great promise in ot her fields of enginee- ness), on viscous and swirl effects, and on buoyancy. Only the exact but intractable ring, it seems worthwhile (and opportune)for the further development of calculation me equations form a mathematical model that accurately describes es under all po hods for turbulent-flow problems in hydraulics to review critically the range of available sible situations. Turbulence models can only give an approximate description, and, with turbulence models with the aim of assess ing their suitability for application to hydraulics a particular set of empirical constants, they are valid only for a certain flow or at the problems. In particular, it is important to find out how well the various models can cope most a range of flows. It is of course desirable in a turbulence model to achieve a good with the manifold complications present in hydraulics problems, such as irregular geome- approximation with a single set of constants far a fairly wide range of flows; only then has tries, buoyancy and free surface effects, It is the purpose of this review to present such a field method incorporating the turbulence model real predictive power. a model for an assessment and to show by way of examples what problems have bean and can be tack- whIch the constants have to be adjusted from flow to flow is in essence little more than a led with the presently available models method for interpolating experimental data, similar to the empirical formulae mentioned above. A good turbulence model should however allow extrapolation from the empirical The number of turbulence models suggest ed in the literature over the last 70 years is rother large so that not all models can be included in this review. Some of the mo- an extrapolation is meaningful dels suggest ed have been tested very little(or not at all) and are therefore difficult to assess; others are used no longer because they have been found to perform poorly The mast universal turbulence model is not necessarily also the most suitable one present review concentrates on those models which are still in use and have been tested for a porticular problem, In practical applications, the economy and ease of use of a model to a fair degree; further the models which have already been applied to hydraulic flow are also important factors, and the more universal models are usually more complex and problems or are specially geared to them will receive speclal attention. Assessment of the thus require more computing time, Thus, for each problem, the right level of complexity applications will focus on free surface flows because flows in closed ducts are not pecu has to be chosen from amongst the available models liar to the field of hydraulics (they are very similar to gas flows occuring in other areas of fLuid mechanics and are therefore covered by the rev iews mentioned above). Futher, 1.2 Scope and Outline of the Review the review concentrates on small-scale(or near-field) issues; for reasons to be explained in detail later, use of the refined models mainly described in this book is not warranted Starting with Prandtl in 1925, researchers have been increasingly active develop for simulating the large-scale horizontal turbulent transport in natural water bodies. Pre PrandtI-mixing -length hypothes isis the most well-known example, related the turbulely a ing turbulence models during the last 20 years. The first turbulence models, of which the vious reviews on turbulence models for hydroulic problems can be found in References [5 to [8]. These prow lde valuable accounts of the state of the art at the time of their ap- transport tems uniquely to local mean-flow quantities. Even these relatively simple mo- pearance, but they are not representative of the present state since significant progress dels could, for a long fime, be used only to calculate self-similar flows(e. g. far field has been achieved in the meantime. In his survey for the 1971 IAHR Congress in Paris, of jets and wakes) for which the partial differential equations expressing mass and mow Biesel [8] expressed the hope that one will learn sufficiently quickly from the imper -4- 5 fections of the existing models in order to soon arrive at models of greater prac- URBULENCE MODELLING tical value it is one of the main tasks of this work to examIne whether this hope has 2 become true in the meantime 2.1 n-Flow Equat ions and the Proolem of Closure The present review is restricted to turbulence models describing the local state of used in integral methods is not discussed here. Further, the numerical rocedure fo solve fow quantities and therefore fom the basis of the so-call ed field methods, The origin of turbulence, such as are employed in field methods, The global descript ion of turbulence these equations are the conservation laws for mass, momentum, thermal energy and spe complete matl ctes entration For incompressible flows, these laws can be expressed in tensor nota model (mean-flow and turbulence equations)are not dicussed either Readers interested fie in numerical procedures in fluid mechanics are referred to the book of roache[? Mass conservation contini uaTion aao n the section to follow(Sec. 2. 1), the problem of calculating furbulent flows is posed more precisely by introducing and discussing the time-averaged equations gov aU ax i (2.1) erning the mean-flow quantities. the appearance of turbulent transport terms in these equations makes apparent the necessity of introducing turbulence models. Before turning ta such models, it is helpful to describe briefly some important features of the turbulent phenomenon, and this is done in Sec. 2.2. In order to review a fairly large number a o Momentum conservation: Novier-Stokes equations different turbulence models, a scheme of classification is needed. This scherne is intro aUi -?r duced in Sec. 2. 3 together with same basic idcas, for example the eddy viscosity con- 22) cept, common to many models, The heart of this book is the actual review of models in Sections 2, 4 to 2.7; the models ore discussed in order of increasing complexity. A se- cond major contribution is contained in Chapter 3, where model appl ications are presenF- Thermal energy /species concentration conservation ed and compared with experimental results, starting with relatively simple problems such as steady channel and jet flows and moving on to increasingly complicated situations, 0+U; a2φ e.g. discharges into stagnant water and info rivers, recirculating and unsteady flows ax: ax 23) Chapter 4 closes the review with an assessment of what the available models can achieve and in whIch areas further research is necessary where Ui is the inst antaneous velocity component in the direction xi, P is the instanta- neous static pressure++) and p is a scalar quantity which may stand for either temper ture T or species concentration c.S is a volumetric source term expressing, for example, heat generation due to chemical or biological react ions. V and 入 are the molecular www.doci Cinematic)viscosity and diffusivity (of f)respectively. Use has been made in the above equat ions of the Boussinesq approximation so that the influence of variable density appears hich is the last term on the right hand side of Eq.( 2.23 involving the refer nsity pr and the gravitational accelerat i in directio Together with an equation of state relat ing the local density p to the local values of T e and c, Ecs.(2. 1)ho( 2. 3)form a closed set and are the exact equations referred to in the Introduction; they therefore describe all the details of the turbulent motion. As was explained in the Introduct ion, these equations cannot at present be solved for turbulent lows of practical relevance. Therefore, a statistical approach is taken and, as suggested by Osborne Reynolds, the instantaneous values of the velocity Ui, the pressure P and the scalar quantity p are separated into mean and fluctuating quantities P=阝+p,φ=+φ」 2.4) A short introduction to tensor notation is given in Appendix a Strictly P is the static pressure minus the hydrostatic pressure at reference density p 【实例截图】
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