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- The LVEL Turbulence Model

Y Y SOLUTN LTLS P P Y P P Y The significances of the settings The meanings of these settings are as follows a ENUT GRND8 This activates the solution for K u and subsequent calculation of the turbulent viscosity b EL1 GRND1 This activates a setting of the length scale to be the distance from the nearest wall taken from the variable WDIS which is computed from LTLS c SOLVE LTLS etc The variable LTLS is solved once only unless the grid is changed during the calculation from its solution the distance of each cell centre from the nearest wall is computed Special settings for user set walls If the user requires a surface to be recognised as a wall which has no other indication of its nature such as a patch of type WALL or the setting of adjacent PRPS or VPOR values a COVAL must be set for the variable LTLS COVAL patch LTLS 1 0 0 0 This is necessary to ensure that the wall is properly recognised during the otherwise automatic distance from the wall calculation C Examples a The labyrinth The following pictures illustrate the application of the LVEL model to the flow in a labyrinthine passage at a low Reynolds number show the distributions of the distance from the wall computed by the scalar equation solver for wall distance and wall gap the effective viscosity which is very small in the low Reynolds number case the temperature with allowance for conduction within the solid and the pressure within the fluid back here The wall distance The effective viscosity The temperature The pressure b Comparison with other models for low Reynolds Number flows The following extracts from a publication by Aganofer Liao and Spalding show how the LVEL model can be used for conjugate heat transfer calculations They show further that the results are as good as those of more widely publicised models which are much more expemsive to run The problem in question is as follows The two steel blocks are uniformly heated The problem is to compute the maximum temperature within the solid at various rates of flow of air adiabatic wall duct steel cavity steel aluminium Fig 1 Horizontal dimension 3 cm vertical dimension 1 2 cm The two steel blocks and the cavity are each 1 cm wide The blocks the aluminium base and the air flow duct are each 0 4 cm thick The models employed Predictions were made with the above indicated input data by activating three of the low Reynolds Number models which are supplied as standard in the PHOENICS computer code namely the LVEL model as described in the present paper the Lam Bremhorst Yap model Lam Bremhorst 1981 Yap 1987 the two layer k epsilon model El Hadidy 1980 Rodi 1991 Since all three of these models require as an input the distance from the wall this was computed by solving the W equation as described above Further details Four different uniform computational grids were

Original URL path: http://www.cham.co.uk/phoenics/d_polis/d_enc/turmod/enc_t312.htm (2016-02-15)

Open archived version from archive - The IMMERSOL model of Radiative Heat Transfer

the expression for λ ref becomes λ eff 16 3 ε s 1 σ T 3 Eq 9 At the other extreme when the medium is so thin as not to participate at all in the radiative heat transfer between two solid sufaces at temperatures T hot and T cold say the heat flux q is well known to obey the formula q 1 1 ε hot ε hot 1 ε cold ε cold 1 σ T hot 4 T cold 4 Eq 10 Equation 8 is of the flux proportional to gradient kind which PHOENICS is well equipped to solve Equation 10 is of the less amenable action at a distance kind The question arises how can the latter be made more like the former b First steps Re writing equation 10 for the case in which T hot T cold is small and in which the wall emissivities are unity yields q 4σ T 3 T hot T cold Eq 11 where T stands for either temperature Since the temperature gradient equals T hot T cold W gap the effective conductivity which corresponds to equation 11 is simply λ eff 4 W gap σ T 3 Eq 12 where W gap stands for the distance between the soild surfaces So the conductivity increases with inter wall distance as it must do if the heat flux is to be independent of that distance It is interesting to compare the value of this conductivity with the thermal conductivities of common materials such as atmospheric air water at 0 degC steel 0 0258 0 569 43 0 wherein the units are W m 1 degC 1 In the same units and with a wall gap equal to one metre the values of λ eff at various temperatures in degrees Celsius are T 3 20 100 500 1000 1500 2000 λ eff 5 706 11 77 104 8 467 9 1264 1 2663 6 Even taking into account that turbulence may increase the effective conductivity of a fluid by two or three orders of magnitude it can be concluded from these tables that radiative heat transfer can be significant at room temperature and at high temperatures it becomes dominant c Between the thick and thin extremes Let now the reciprocal of conductivity be considered i e the resistivity measured in degC m W For the thick medium eq 9 yields λ eff 1 3 16 ε s σ T 3 Eq 13 and for the thinnest possible totally empty medium eq 12 yields λ eff 1 1 4 W gap σ T 3 Eq 14 It is therefiore not unreasonable to suppose that for intermediate conditions the two multipliers of 4σ T 3 should be added so as to cr a more generally valid single resistivity formula thus λ eff 1 3 4 ε s 1 W gap 4σ T 3 Eq 15 This is the source of equation 7 introduced in section 3 1 d above d Wall emissivity as an extra resistance Equation 7 is used for the calculation of the T 3 diffusion fluxes of the finite volume equations which PHOENICS solves but something special is done for coefficients when one node lies in the transparent medium and the other within a solid as exemplified by nodes B and A in the following figure in which for simplicity the transparent medium consists of a single phase As is usual in PHOENICS the conductivities pertaining to the cell are stored at each grid node Therefore the radiative heat flux crossing the boundary between cell B and cell C will be deduced from the formula flux B to C T 3 B T 3 C x I x B λ 3 B x C x I λ 3 C where x is the horizontal co ordinate However the calculation of the radiant flux as the S interface requires more careful study because the surface emissivity can cause a discontinuity of T 3 gradient there as is illustrated in the following figure which shows the postulated profiles of both T 3 and T 1 because of their inescapable interaction Here it is postulated that T 3 and T 1 are equal to each other within the solid but whereas the latter has a finite gradient everywhere the latter may have an infinite one at the interface between the phases The fluxes of energy in question are as follows conduction from A to S namely T 1 A T 1 S λ 1 A x S x A conduction and convection from S to B namely T 1 S T 1 B λ 1 B x C x S radiation from S to B namely T 3 S T 3 B x C x S λ 3 B 1 ε S ε S wherein the term involving ε S is inserted so as to conform with equation 10 above Requiring them to be in balance at the surface S enables the equal by definition values of T 3 and T 1 there to be evaluated The necessary formula is as follows T 3 S T 3 S T 1 A λ 1 A x S x A T 1 B λ 1 B x B x S T 3 B x B x S λ 3 B 1 ε S ε S λ 1 A x S x A λ 1 B x B x S 1 x B x S λ 3 B 1 ε S ε S Eq 16 3 3 How W gap is calculated Click here for a full explanation 3 4 Implementaton in PHOENICS a Activation via Q1 Few actions are needed in order to activate IMMERSOL They comprise SOLVE T3 to ensure that T3 is solved DISWAL to activate the WDIS and WGAP calculation STORE WGAP to enable this quantity to be accessed together with such information as provides material properties and initial and boundary conditions Warning It is also desirable to set varmax tem1 and varmax

Original URL path: http://www.cham.co.uk/phoenics/d_polis/d_enc/enc_rad3.htm (2016-02-15)

Open archived version from archive - DISTANCE from the WALL (WDIS), and between walls (WGAP)

feature 2 Examples The next three pictures represent a computational grid in a curved duct joining two box shaped spaces the distance from the wall computed by the procedure to be described the distance between the walls computed by the same procedure Their inspection may lend credibility to the claims of plausibility which have just been made A relevant core library case Case 119 of the core library concerns the distribution of LTLS WDIS and WGAP within an enclosure which optionally contains a solid It is a convenient starting point for newcomers to the subject 3 The differential equation for L The LTLS feature involves the solution for a scalar variable L which obeys the differential equation div grad L 1 within the fluid and equals zero within solid materials and at thin surfaces to which no slip boundary conditions are applied to the velocity equations This equation is similar to that for temperature within a uniformly conducting medium having a uniform heat source and in contact with solids and other surfaces at which the temperature is held at zero It is of course easily solved by the linear equation solver of PHOENICS whether the geometry is one two or three dimensional This solution needs to be determined once only if the solids do not change position 4 Deduction of the distance from and between walls The variable L is not itself the distance from the wall even though it is proportional to that distance at locations which are near a wall Its dimensions are indeed those of length squared However that distance can be deduced from the solution for L as can also a plausible estimate of the effective distance between walls The method is to derive a relationship between the distance from the wall W dis and the distance between walls W gap on the one hand and the local value of L and the local value of its gradient on the other from consideration of a simple geometry namely that between two parallel walls and then to presume that it has general validity 5 The parallel wall situation Let the distance measured from one wall be y and the distance to the opposite wall y 1 The differential equation takes the one dimensional form d 2 L d y 2 1 which can be integrated to give d L dy y A where A is a constant and then further to L y 2 2 Ay B where B is another constant Insertion of the boundary condition L 0 at y 0 and y y 1 yields B 0 and A y 1 2 with the result L y y 1 y 2 L y 1 2 y where L stands for d L d y Elimination of y 1 from the first equation by use of the second yields L y L y 2 which is an easily soluble quadratic equation From its solution follow with W dis substituted for y and W gap for y

Original URL path: http://www.cham.co.uk/phoenics/d_polis/d_enc/dist.htm (2016-02-15)

Open archived version from archive - Turbulence models for CFD in the 21st century; Example 1

thermally induced volumetric expansion of the solid The temperature distribution in the solid and fluid when heat is supplied along the axis and gravity acts vertically The associated velocity vectors in the fluid and displacement vectors in the solid The thermally induced radial stresses in the solid The thermally induced circumferential stresses in the solid The computed distance from the walls needed by the underlying LVEL model of the horseshoe

Original URL path: http://www.cham.co.uk/phoenics/d_polis/d_lecs/turb2000/examp001.htm (2016-02-15)

Open archived version from archive - Turbulence models for CFD in the 21st century; Example 2

falls later as heat is transferred to the fresh water fluid This picture shows how the fresh water temperature rises gaining its heat from the saline fluid Here is shown the density of the saline fluid It first diminishes because of the temperature rise eventually becoming lighter than the fresh water As it cools it becomes heavier again Here is shown the density of the fresh water fluid This also

Original URL path: http://www.cham.co.uk/phoenics/d_polis/d_lecs/turb2000/examp002.htm (2016-02-15)

Open archived version from archive - Turbulence models for CFD in the 21st century; Example 4

primary secondary and dilution air streams The graphical convergence monitor for the 40 fluid run shown here gives proof of a satisfactorily converging calculation and computer times are seen to be small The results The table shows how the number of fluids influences the predicted rate of smoke production and the computer time number smoke seconds 1 0 74 100 10 2 38 139 20 2 28 217 30 2 31 267 40 2 26 485 50 2 27 599 Note that On this occasion MFM predicts more smoke production than the conventional single fluid model and the 10 fluid model provides a good approximation The following figures show the computed PDFs for a location in the middle of the outlet plane of the combustor for 10 fluids 40 fluids 50 fluids The shapes are all similar and the root mean square and population average values do not differ much The following contour plots show various aspects of the 50 fluid calculation The very different smoke distributions on an axial plane according to a the single fluid no fluctuations model and b the multi fluid model The flow is from right to left The somewhat different distributions of population average

Original URL path: http://www.cham.co.uk/phoenics/d_polis/d_lecs/turb2000/examp004.htm (2016-02-15)

Open archived version from archive - Turbulence models for CFD in the 21st century; Example 3

in a gas turbine combustor Predictions of the smoke production rate have been made with various population grid finenesses Some results are contained in the following table number of fluids production rate comments 1 6 09 much too high 3 1 22 much too low 5 3 65 getting close 10 3 99 still varying 20 3 32 very close 100 3 23 presumed correct Since computer time increases with

Original URL path: http://www.cham.co.uk/phoenics/d_polis/d_lecs/turb2000/examp003.htm (2016-02-15)

Open archived version from archive - Turbulence models for CFD in the 21st century; Example 5

The sketch illustrates the apparatus and the initial state of the two liquids They are both at rest and are separated by a horizontal interface The paddle is supposed to be suddenly set in motion The computational task is to predict both the macro mixing represented by the subsequent distributions of time average pressure velocity and concentration and the micro mixing ie the extent to which the two liquids are mixed together at any point d The turbulence model combination of MFM and k epsilon There is no need simply because MFM is available to refrain entirely from using conventional models In the present study therefore the k epsilon model was used as the source of the length scale effective viscosity and micro mixing information This allowed the power of the MFM to be concentrated on the process which it alone can simulate namely micro mixing and the subsequent chemical reaction An eleven fluid model was employed with mass fraction of material from the top half of the reactor ie the acid as the population distinguishing attribute This was probably not sufficient to permit achievement of grid independent of solutions However one of the merits of MFM is that population grid refinement studies are easily performed e The fluid population distributions The problem can be regarded as a five dimensional one for it has three space dimensions one time dimension and one population dimension The simulation procedure therefore generates a very large volume of data of which only a tiny fraction can be presented here What will first be shown is a series of PDFs They all pertain to a single instant of time namely that at which the paddle has revolved ten times a single vertical position namely about one third of the height from the base and positions on a single radial line Specifically PDFs will be shown for six radial locations starting near the axis and moving outward to about one third of the radius Near the axis most of the fluid is still alkaline Farther from the axis more acidic fluids are found This tendency increases with increase of radius Also the shape of the PDF changes The shape change continues This is the last PDF to be shown f The salt distribution after 10 paddle rotations The acid and the alkali are supposed to react chemically in accordance with the classic prescription acid base salt water Of course no chemical reaction can occur in the unmixed fluids which to use the parent offspring analogy are the Adam and Eve of the whole population It is only their descendants with both acid and alkali in their blood who can produce any salt This is why the prediction of the yield of salt necessitates knowledge of the concentrations of the offspring fluids The next picture shows the salt concentrations after 10 paddle rotations which the multi fluid model has predicted These concentrations are the averages for all eleven fluids The salt concentrations predicted by the multi fluid

Original URL path: http://www.cham.co.uk/phoenics/d_polis/d_lecs/turb2000/examp005.htm (2016-02-15)

Open archived version from archive

web-archive-uk.com, 2016-10-21