a general one-equation turbulence model for free shear and wall-bounded flows

Flow, Turbulence and Combustion 73: 187–215, 2004.C© 2004 Kluwer Academic Publishers. Printed in the Netherlands. 187

A General One-Equation Turbulence Model for FreeShear and Wall-Bounded Flows

EHAB FARES and WOLFGANG SCHRODERAerodynamisches Institut, RWTH Aachen Wullnerstrasse 5-7, Aachen, Germany;E-mail: [email protected]

Received 17 June 2003; accepted in revised form 6 April 2004

Abstract. The purpose of this work is to introduce a complete and general one-equation model capableof correctly predicting a wide class of fundamental turbulent flows like boundary layer, wake, jet, andvortical flows. The starting point is the mature and validated two-equation k-ω turbulence model ofWilcox. The newly derived one-equation model has several advantages and yields better predictionsthan the Spalart–Allmaras model for jet and vortical flows while retaining the same efficiency andquality of the results for near-wall turbulent flows without using a wall distance. The derivation andvalidation of the new model using findings computed by the Spalart–Allmaras and the k-ω modelsare presented and discussed for several free shear and wall-bounded flows.

Key words: one-equation, turbulence, model

1. Introduction

Many turbulence models were introduced in the past ranging from simple and in-complete algebraic models like the mixing length hypothesis by Prandtl [38] tocomplete stress-transport models like the Launder–Reece–Rodi model [20]. How-ever, the increasing complexity of the turbulence models and the larger computa-tional effort, when they are applied to realistic technical problems, are not alwaysjustified by a qualitative improvement in the solutions. An overview on some of thestandard and widely used turbulence models is given in [62].

One-equation turbulence models enjoyed a wide popularity in the last decade.The growing interest in this type of models is explained primarily by the numericalease of use compared to standard two-equation models like the k-ε [18] and k-ωmodels [61]. Algebraic models like the customary Baldwin–Lomax model [1] areefficient from a numerical point of view but lack generality and have deficits likethe missing transport and diffusion effects. Comparatively, one-equation modelsrepresent the simplest form of turbulence models that include transport effectsand are considered to be a good compromise between the algebraic and classicaltwo-equation models.

The first one-equation model for the turbulent kinetic energy k was postulatedby Prandtl [39] and independently introduced by Emmons [9]. Both formulationswere incomplete since they needed the definition of a problem dependent length

188 E. FARES AND W. SCHRODER

scale. Later Bradshaw, Ferris, and Atwell [6] proposed another model for k basedon the assumption that the Reynolds shear stress in a 2D flow τtxy is proportionalto the turbulent kinetic energy k

τtxy = βb k with βb = 0.3 (1)

which is known as the Bradshaw hypothesis and was validated by Townsend [57]through numerous measurements in boundary layers, wakes and mixing layers.

The first formulation for a one-equation model directly for νt was postulatedby Nee and Kovasznay [33] based on phenomenological arguments. A completemodel for νt was also proposed by Secundov [50] as an extension of the formerNee and Kovasznay model with considerable success. Its current formulation, whichwas unknown in the English speaking community until its publication in 1995 iscalled the νt92 version [51]. In 1991 Baldwin and Barth [2] derived an elaboratetransport model from the classical k-ε model, which had limited success due toits inconsistent numerical behavior at turbulent/nonturbulent interfaces [28, 62].The most successful one-equation model up to date was introduced in 1992 bySpalart and Allmaras [52]. It was developed generically for the turbulent eddy vis-cosity νt with strong emphasis on the numerical behavior. Later Menter [28] pro-posed a general methodology for deriving one-equation models from two-equationmodels and suggested a new model based on the standard k-ε equations, whichshows superior results when compared to the Baldwin–Barth model. Nagano, Pei,and Hattori [32] also derived another one-equation model from the low-Reynolds-number k-ε model with many closure and damping functions and showed goodresults compared with DNS and experiments for wall-bounded flows like bound-ary layers and plane wall jets but did not discuss any aerodynamic or free shearflows.

The performance and the predictive capabilities of many models including theBaldwin–Lomax, Baldwin–Barth, Spalart–Allmaras, standard k-ε, k-ω, and theSST [27] models are documented in many publications [3, 13, 42, 43, 46, 51, 59, 62]and corroborate the quality of the one-equation models, especially the Spalart–Allmaras model. There are, however, some limitations in the generality of almostall one-equation models, since they lack a correct prediction of both wall-boundedand free shear flows simultaneously. This drawback is eliminated in the proposedturbulence model, while maintaining the same efficiency of other one-equationmodels.

First, the model is derived mathematically from the two-equation k-ω model.Closure functions and coefficients are stipulated with the help of relevant turbulentflows. In the second part the final version of the model is given and comparedwith the Spalart–Allmaras model. The third part constitutes the detailed valida-tion and discussion of the model based on wake, jet, vortical, wall-bounded, andaerodynamic flows. Finally, a short summary is given.

A GENERAL ONE-EQUATION TURBULENCE MODEL 189

2. Derivation

Many technically relevant complex flows are composed of fundamental flows likeboundary layer, wake, jet, and vortical flows. The quality of the overall solution isdetermined by the capability to capture the detailed physics of these flows whichis why the new model focuses on the prediction of such basic phenomena.

The starting point will be the mature and validated two-equation k-ω turbulencemodel of Wilcox [62] given in tensor notation

Dk

Dt= τti j

∂ui

∂x j− β∗kω + ∂

∂x j

[(ν + σ ∗νt )

∂k

∂x j

](2)

Dω

Dt= α

ω

kτti j

∂ui

∂x j− βω2 + ∂

∂x j

[(ν + σνt )

∂ω

∂x j

](3)

with the following closure coefficients

σ ∗ = σ = 0.5 , α = 0.52 , β∗c = 0.09 , βc = 0.072 (4)

and the closure functions

β∗ = β∗c f ∗

β f ∗β = 1 + 680ψ2

k

1 + 400ψ2k

ψk = max

(0.,

1

ω3

∂k

∂x j

∂ω

∂x j

)(5)

β = βc fβ fβ = 1 + 70ψω

1 + 80ψω

ψω =∣∣∣∣i j jk Ski

(β∗c ω)3

∣∣∣∣ (6)

using the definitions

i j = 1

2

(∂ui

∂x j− ∂u j

∂xi

)Si j = 1

2

(∂ui

∂x j+ ∂u j

∂xi

)(7)

νt = k

wτti j = 2νt Si j . (8)

The functions ψk and ψω were added to the original model [61] to describecross-diffusion and vortex-stretching, respectively [62]. They mainly improve theresults for wake and jet flows without interfering with the good results achievedwith the original model for boundary layer flows. Note that the k-ω model is oneof the few turbulence models, which do not need any viscous damping or possessany explicit wall distance dependence in the equations. However, numerical dif-ficulties are encountered in the vicinity of the wall, since the boundary conditionωwall → ∞ produces high gradients near the wall and requires a higher resolutionin the boundary layer. Further criticism [26, 27] concerning the freestream valueωinit and ωinflow sensitivity are partly remedied in the new formulation of the k-ωmodel [62]. Such criticism neglects the fact that the boundary values of a turbulencemodel like in any other boundary value problem should affect the solution, e.g., inthe order of the free stream value, a fact that is rarely mentioned in the debate.


An equation for νt can be derived using the definition of the eddy viscosityk = νtω and the total derivatives Dk/Dt and Dω/Dt [2, 28]:

Dνt

Dt= 1

ω

(Dk

Dt− νt

Dω

Dt

)= (1 − α)

τti j

ω

∂ui

∂x j− β∗k + βνtω

+ 1

ω

∂

∂x j

[(ν + σ ∗νt )

∂k

∂x j

]− νt

ω

∂

∂x j

[(ν + σνt )

∂ω

∂x j

]. (9)

The terms on the right-hand side are rewritten1 by inserting k = νtω and usingthe equivalence σ ∗ = σ . The diffusion terms on the right-hand side is reformulatedequivalently yielding the νt equation

Dνt

Dt= 2(1 − α)

νt Si j

ω

∂ui

∂x j− (β∗ − β)νtω + ∂

∂x j

[(ν + σνt )

∂νt

∂x j

]

+2(ν + σνt )

ω

∂νt

∂x j

∂ω

∂x j(10)

where ψk is redefined

ψk = max

[0.,

1

ω3

(νt

(∂ω

∂x j

)2

+ ω∂νt

∂x j

∂ω

∂x j

)]. (11)

Note that since so far only a reformulation without any simplification has beencarried out there is no loss in generality compared to the k-ω formulation in Equa-tions (2, 3). The division by ω in two terms on the right-hand side is no problem ifω → 0 since this is physically only possible in laminar flows νt = 0, which meansthat the numerator also tends to zero. Usually a max(ω, ε) with a small numericalvalue ε, e.g., the machine accuracy, is used to numerically prevent division by zero.The value ω should be reconstructed explicitly based on flow quantities such asvelocity gradients. By defining this reconstruction we close our proposed equation.It is clear that this is the most essential step, since the generality of the reconstructedω must be guaranteed throughout a wide range of flows.

The most appropriate assumption concerning such a reconstruction is theBradshaw hypothesis given in Equation (1). This assumption is directly imple-mented into many turbulence models like in [6, 28] and is indirectly included inseveral other turbulence models like k-ε [18]2 and k-ω [62]. For instance, neglect-ing the convective and dissipative terms of the turbulence energy Equation (2) theaforementioned hypothesis can be easily derived

β∗kω = β∗c k2/νt = τti j

∂ui

∂x j⇒ k = 1√

β∗c

√νtτti j

∂ui

∂x j(12)

and for 2D shear layers

k = 1√β∗

c

τtxy = νt√β∗

c

∣∣∣∣du

dy

∣∣∣∣ . (13)


DNS and experimental data indicate that this hypothesis is neither exactly validin the viscous sublayer of the turbulent boundary layer [32] nor in free shear lay-ers [48]. Several attempts based on dimensional analysis and on simplificationsof the k-ω equations were studied to achieve other reconstructions. However, theyusually lead to ill numerical behavior, since they include higher-order derivativesand cross derivatives of u and νt and very strong gradients near the wall similarto the underlying k-ω model. On the one hand, an exact reconstruction means aninfinite value of ωwall → ∞ at the wall, which on the other hand should be avoidedfor the sake of efficiency and stability of the numerical scheme. Consequently, aslight modification of the coefficients of Equation (10) and its closure functionsis preferred in the formulation of the new one-equation turbulence model. Thequantity ω is reconstructed using Bradshaw’s hypothesis for 2D shear layers

ω = k

νt= 1√

β∗c

∣∣∣∣du

dy

∣∣∣∣ (14)

and generalized using the norm of the strain tensor [32]

ω = 1√β∗

c

√2Si j Si j . (15)

In the next sections several tuning and closure coefficients and functions are pre-sented, which were evaluated using computations of free shear and wall boundarylayer flows.

2.1. SHEAR EDGE

The value of σ = 0.5 may lead to a non-consistent behavior of the model atthe turbulent/nonturbulent edge like the one observed with the Baldwin–Barthmodel [28, 54, 62] and a higher value is suggested. A recalibration of theconstants σ and α of the model especially for wall bounded flows led to thevalues

σ = 1.2 and α = 0.29. (16)

2.2. FREE SHEAR FLOWS

The calibration of the closure function fβ∗ for jet flows leads to the followingadopted formulation

f ∗β = 1 + 650ψk

1 + 100ψk. (17)

A recalibration was necessary to achieve the same good agreement between cal-culations using the k-ω model and experiments. Note, that Wilcox introduced thisfunction and calibrated it to improve the original k-ω model for the same free shearflows. This function does not play a role in the vicinity of the wall [62] since ψk

approaches zero near solid surfaces.


2.3. NEAR-WALL BEHAVIOR

The Bradshaw hypothesis is exactly fulfilled in the logarithmic region of the bound-ary layer. This probably explains the success of this assumption in the turbulencemodeling community. However, it is not valid in the viscous sublayer, where thelaminar viscosity is dominating. Almost all turbulence models include dampingterms near the wall to correct this behavior. Many of these terms are functionsof the wall normal distance d. To avoid such a dependence the viscous dampingfunction fv1 of Mellor and Herring [25] with a calibrated value of cv1 similar to theapproach applied in the Spalart–Allmaras model was chosen

νt = ν fv1 with fv1 = χ3

χ3 + c3v1

χ = ν

νcv1 = 9.1. (18)

This approach allows ν to be equal to κuτ d in the logarithmic region and in theviscous sublayer. This behavior is considered one of the reasons for the success ofthe Spalart–Allmaras model, since it enforces a numerically advantageous lineargrowth of ν in the vicinity of the wall. Recall that fv1 is important only in the viscousregion, where χ is of O(1), while its influence disappears in the logarithmic region.For free shear flows it has a limited impact at turbulent/nonturbulent edges, whereχ is also of O(1). In these regions we expect the turbulent eddy viscosity νt to beof small influence and therefore not to interfere with the former calibration for freeshear flows.

There are investigations of the correct wall limiting behavior of νt (d → 0) [32,62] that will not be discussed here, since in the viscous sublayer the laminar viscosityν is the dominating quantity. Hence, good results can be expected for wall-boundedflows even if νt is not asymptotically consistent at the wall.

2.4. DECAY OF ISOTROPIC TURBULENCE

Experimental investigations [57] indicate that the turbulent kinetic energy k of anisotropic homogeneous turbulent flow should decay according to

k(t) ∼ t−n with n = 1.25 ± 0.06 (19)

The k-ω model can be simplified by dropping all the spatial gradientsdk

dt= −β∗

c ωk anddω

dt= −βcω

2 (20)

and solved analytically for large t

k(t) ∼ t−β∗c /βc = t−5/4 and ω(t) ∼ 1

βct(21)

which is in good agreement with experiments. Similar to other one-equation modelsthe proposed one-equation model in Equation (10) does not predict a decay at all,


since all terms include spatial derivatives that vanish in a homogeneous flow leadingto dνt/dt ≡ 0. The correct decay of νt can be derived from the k-ω equation

νt (t) = k(t)/ω(t) ⇒ νt (t) ∼ t1−β∗c /βc = t−1/4 (22)

To achieve this behavior a generic term is added to Equation (10)

dνt

dt= −kinit

(φ

νt

ν

) −β∗c /βc

1−β∗c /βc = −kinit

(φ

νt

νinit

)5

(23)

where kinit represents the initial turbulent kinetic energy that can be calculated fromthe initially, i.e. prescribed turbulence intensity Tu

Tu =√

2k

3u2∞. (24)

Furthermore, the following relation can be derived using Equations (20) for t = tinit

dνt

dt

∣∣∣∣init

= 1

ω

(dk

dt− k

ω

dω

dt

)∣∣∣∣init

= (βc − β∗c ) kinit = −kinit

(φ

νt

νinit

)5

(25)

This leads to the value of

φ = (β∗c − βc)1/5 ≈ 0.4478. (26)

The small value of φ < 1 taken to the power of 5 does not interfere with theformer calibration of the model. Furthermore it should be noted that this additionalterm is not essential in the calculation of turbulent steady flows.

A simulation of isotropic turbulence using the original k-ω model and the newmodel with the additional term given in Equation (23) is presented in Figure 1. Itshows the excellent agreement between the proposed term and the predicted decayof the original model.

3. Final Version of the Model

Using

νt = ν fv1 with fv1 = χ3

χ3 + c3v1

and χ = ν

ν(27)

the new one-equation turbulence model reads for the dependent variable ν

Dν

Dt= 2(1 − α)

ν

ωSi j

∂ui

∂x j− (β∗ − β)νω − kinit

(φ

ν

νinit

)5

+ ∂

∂x j

[(ν + σ ν)

∂ν

∂x j

]+ 2

(ν + σ ν)

ω

∂ν

∂x j

∂ω

∂x j(28)


Figure 1. Decay of isotropic turbulence.

with the closure coefficients

σ = 1.2 α = 0.29 β∗c = 0.09

βc = 0.072 cv1 = 9.1 φ = (β∗

c − βc)1/5

(29)

and the closure functions

β∗ = β∗c f ∗

β f ∗β = 1 + 650ψk

1 + 100ψk

ψk = max

[0.,

1

ω3

(ν

(∂ω

∂x j

)2

+ ω∂ν

∂x j

∂ω

∂x j

)](30)

β = βc fβ fβ = 1 + 70ψω

1 + 80ψω

ψω =∣∣∣∣i j jk Ski

(β∗0 ω)3

∣∣∣∣ ω = 1√β∗

c

√2Si j Si j . (31)

The initial and boundary conditions for ν

νinit ≈ 0.1ν νwall = 0 νinflow = νinit∂ν

∂n∣∣∣∣outflow

= 0 (32)

with the initial turbulent kinetic energy kinit = 3/2 u2∞Tu2|init.

There are many similarities between the Spalart–Allmaras (S A) model and thenew one-equation model like the same transformation from νt to ν and the same


Table I. Comparison between the right-hand-side terms of the Spalart–Allmaras andthe new model.

Term S A model New model

Production cb1ν

(√2i ji j + ν

κ2d2

)2(1 − α)ν

(Si j

ω

∂ui

∂x j− 1

3δi j

∂ui

∂x j

)

Destruction cw1 fw

(ν

d

)2

(β∗ − β)νω

Diffusion 11

σ

∂

∂x j

[(ν + ν)

∂ν

∂x j

]∂

∂x j

[(ν + σ ν)

∂ν

∂x j

]

Diff./Dest.cb2

σ

∂ν

∂x j

∂ν

∂x j2

(ν + σ ν)

ω

∂ν

∂x j

∂ω

∂x j

Decay – kinit

(φ

ν

νinit

)5

convective term. In Table I the right-hand side terms of both turbulence modelsare juxtaposed. Turbulence is produced in the S A model mainly through vorticity,whereas in the new model it is produced through shear stress. For a 2D free shearlayer the production term in both models reduces to ∼∂u/∂y but with differentproportionality factors. The destruction terms are completely different, since thereis no dependence on the wall distance d in the new model. The new model possessesa more difficult formulation of the destruction term determined by the closurefunctions f ∗

β , fβ, ψk, ψω. The first diffusion term is almost identical in both modelsexcept for the coefficients. A second diffusion term depends only on the gradientof ν in the S A model but in the new model is a function of the gradient of ω. Acareful investigation of the term 2 (ν+σ ν)

ω∂ν∂x j

∂ω∂x j

shows that this term is almost alwaysnegative, i.e., this term acts as a destruction term. Unlike in the S A model thequantity ω reconstructed through shear stress plays a major role in the production,destruction, and diffusion terms. Finally, the new model is extended to account forthe decay of turbulence, a feature that is missing in the Spalart-Allmaras model.

4. Validation and Discussion

The Favre-averaged Navier–Stokes equation, and the equation of the turbulencemodel are solved numerically for several flow problems. Boundary layer solu-tions are achieved using similarity transformations as described in [62]. 2D flowsare calculated on the basis of a finite-volume discretization on block-structuredgrids. The viscous stresses were discretized using standard second-order accuratecentral schemes and the spatial discretization of the convective terms follows theAUSM+ scheme [23] using the second-order MUSCL-interpolation [21]. The resultsachieved with the proposed new turbulence model are compared with experimentalresults and with data of the k-ω and the Spalart–Allmaras model. First, examples


Figure 2. Comparison of computed and measured velocity profiles for the mixing layer. Mea-surements by Liepmann and Laufer [22].

of free shear flows such as the mixing layer, wake, and jets are presented followedby the simulation of a turbulent vortex. Finally, the wall bounded turbulent flowslike the flat plate with and without pressure gradients as well as the flow over oneand three-element airfoils are discussed.

4.1. TURBULENT FREE SHEAR FLOWS

The velocity distributions of the incompressible mixing layer are depicted inFigure 2. All three turbulence models give likewise accurate predictions of theflow.

The incompressible far wake flow characterized by a velocity deficit and encoun-tered e.g. in aerodynamical flows behind the trailing edge of wings is presented inFigure 3. The three models give similar results for the core region but deviate atthe edge of the shear layer. The k-ω model shows a much smoother approach ofthe velocity profile at the edge than the Spalart–Allmaras and the new model whencompared to the measurements [10, 60].

Next, two types of incompressible jets are simulated. The plane jet is shownin Figure 4 and the round axisymmetric jet is given in Figure 5. The Spalart–Allmaras model performs poorly in both cases. This model has been optimizedfor aerodynamic applications, which include wake flows and mixing layers but jetflows were not considered. The missing destruction term of the original Spalart–Allmaras model far away from the wall is responsible for this behavior. The new


Figure 3. Comparison of computed and measured velocity profiles for the far wake. Measure-ments (1) by Fage and Falkner [10], measurements (2) by Weygandt and Mehta [60].

Figure 4. Comparison of computed and measured velocity profiles for the plane jet. Measure-ments (1) by Bradbury [5], measurements (2) by Heskestad [16].


Figure 5. Comparison of computed and measured velocity profiles for the round axisymmetricjet. Measurements (1) by Wygnanski and Fiedler [63], measurements (2) by Rodi [41].

model gives the best prediction in the core and edge region of the plane jet andthe quality of the findings is comparable with that of the k-ω model in the middleregion.

The results for the round axisymmetric jet in Figure 5 show a dramatic dis-crepancy of the distribution of the Spalart–Allmaras model. In a wide range itoverpredicts the velocities and the spreading rate by a factor greater than 2.The k-ω and the new model yield almost equally good velocity distributions.However, especially in the outer region the comparison with the experimentaldata evidences a slight superiority of the new one-equation model over the k-ωformulation.

The spreading rate is generally defined as the value of the similarity variableη = y/x where the velocity or velocity deficit is half its maximum value. For themixing layer the spreading rate is defined as the difference between the values of η

where U 2/U 21 is 0.9 and 0.1.

The spreading rates given in Table II provide a concise criterion of the pre-dictive capabilities of the turbulence models for free shear layers and confirmthe quality of the new model. Good agreement with the experimental findings ofthe mixing layer and the wake is achieved by the new model. Furthermore, al-most all popular turbulence models including the S A model predict a strongerspreading of the round axisymmetric jet than the plane jet, which contradicts theexperimental results given in Table II. This phenomenon is known as the round/planejet anomaly and is discussed at length in [37, 62]. The k-ω model and the new


Table II. Spreading rates for turbulent free shear flows.

Flow k-ω model S A model New model Measured

Mixing Layer 0.106 0.105 0.123 0.115 [22]

Far Wake 0.339 0.341 0.336 0.365 [10]

Plane Jet 0.100 0.155 0.111 0.100 – 0.110 [41, 63]

Round Jet 0.088 0.251 0.094 0.086 – 0.096 [5, 16]

one-equation model, however, yield the proper tendency as to the spreading ratefor both jet flows.

It should be noted that no special freestream sensitivity was found for the newlyproposed model as will be described later.

4.2. VORTEX IN TURBULENT FLOW

The behavior of a longitudinal axisymmetric slender vortex in turbulent flow repre-sents a simplified example of a wingtip vortex. Assuming that the mean propertiesof the flow are independent of the axial coordinate the incompressible Reynoldsaveraged equations in polar coordinates read for the azimuthal velocity vt and theradial velocity ur in the radial coordinate r

∂vt

∂t= 1

ρ

[1

r2

∂(r2τ )

∂r− 1

r2

∂

∂rr2v′

t u′r

](33)

with τ = µr∂

∂r

(vt

r

). (34)

The Reynolds stress is modeled according to the Boussinesq hypothesis

v′t u′

r = µtr∂

∂r

(vt

r

), (35)

which leads to the following tangential momentum equation

∂vt

∂t= (ν + νt )

(∂2vt

∂r2+ 1

r

∂vt

∂r− vt

r2

)+

(∂vt

∂r− vt

r

)∂νt

∂r. (36)

Using the circulation � = 2πrvt Equation (36) yields

∂�

∂t= (ν + νt )

(∂2�

∂r2− 1

r

∂�

∂r

)+

(∂�

∂r− 2

�

r

)∂νt

∂r. (37)

Experimental [17, 36], theoretical [14, 35, 45], LES [56], and DNS [34, 40] datashow that self-similarity exists with the similarity variable η = r/r1 and r1 ∼ t1/2,where the subscript 1 denotes the position of maximum tangential velocity. Hoffmanand Joubert [17] proposed that a triple layer structure of the turbulent vortex coreexists obeying different laws for the ratio of circulation �/�1

Visc. layer: 0 ≤ η < 1�

�1= Cvη

2 (38)


Log. layer: η ≈ 1�

�1= 1 + ln η (39)

Outer layer: η � 1�

�1= �∞

�1= const. > 1 (40)

with the constant Cv = 1.83 determined experimentally.3 Furthermore, it wasshown [14, 44] that the integral value of circulation deficit yields∫ ∞

0

(1 − �

�∞

)η dη ∼ 2ν

�∞→ 0 for

ν

�∞→ 0 (41)

at high Reynolds numbers. Given the similarity laws of the turbulent vortex inEquations (38–40) and the ratio �

�∞< 1.0 for the viscous layer it can be deduced

that there must be a region where ��∞

> 1.0 to fulfill the aforementioned integralconstraint. This means that an overshoot in the circulation distribution must exist.The overshoot is a consequence of the self-similar turbulent vortex decaying fasterthan the laminar Lamb–Oseen vortex as discussed in [45]. DNS simulations [34, 40]confirm an overshoot of approximately 2–3% in a turbulent vortex. Calculations byZeman [64] using the k-ε turbulence model and the Reynolds stress closure modelof Durbin [8] show that the correct decay and hence, the correct slight overshootcan only be predicted by the complex Reynolds stress model, whereas the standardk-ε model gives overshoots of more than 50%. Similar results were also found byGrasso et al. [15] who investigated the compressibility effects on the behavior ofthe turbulent vortex.

Equation (36) or (37) can be integrated numerically using the analytic solutionof the laminar Lamb–Oseen vortex described in [4, 45] as initial distribution. Thenumerical solution using the Spalart–Allmaras model in its original version plus therotation and curvature corrections described in [53]4 and the new turbulence modelare compared with the similarity solutions in Figure 6. All solutions generatedfor the Reynolds number Re� = �init/ν = 1000 at the dimensionless time T =t v1init/r1init = 1000 give good agreement in the viscous and logarithmic region,whereas the circulation overshoot presented in Figure 7 deviates strongly dependingon the turbulence model used. The new model gives an overshoot of 2% which isin good agreement with DNS simulations. The Spalart–Allmaras model gives amuch stronger overshoot of more than 20%, a value which is almost unaffected bythe rotation correction term. The temporal development of the maximum tangentialvelocity v1 presented in Figure 8 shows a much stronger decay of the vortex for bothSpalart–Allmaras model versions than for the new turbulence model. The slopeof the decay t−1/2 for the self-similar turbulent vortex discussed in [15, 45, 64]is approximately predicted by the new model and the Spalart–Allmaras modelusing the rotation and curvature corrections. However, the Spalart–Allmaras modelapproaches the slope at much smaller values of the maximum tangential velocitythan the newly developed model.


Figure 6. Comparison of numerical and analytical solutions of the self-similar turbulent vortexfor Re� = 1000 at T = 1000 using different turbulence models.

Figure 7. Comparison of the distribution of �/�∞ for Re� = 1000 at T = 1000 evidencingthe circulation overshoot using different turbulence models.


Figure 8. Comparison of the temporal development of the maximum tangential velocity v1 forRe� = 1000 using different turbulence models compared with the self-similar decay which isproportional to ∼ t−1/2.

It can be concluded that the new model gives the best prediction of the circulationovershoot and the decay of the turbulent vortex. The findings of the new model are inexcellent agreement with calculations using complex Reynolds stress models [64]and DNS [34, 40].

4.3. TURBULENT FLAT PLATE BOUNDARY LAYER

A detailed investigation of the plane incompressible turbulent Couette flow usingdimensional analysis, estimating the order of molecular and turbulent momentumtransfer and the subdivision of the flow into two layers led to the universal law ofthe wall [12, 48]. Relations u+(y+) for the different regions within the turbulentboundary layer can be found in [12, 31, 55]. Based on the Morkovin hypothesis [30]for small Mach numbers Ma∞ < 5 and according to the analysis of Van Driest [58]these results are also valid for compressible flows with minor modifications. Fol-lowing [48, 58] the skin-friction coefficient c f for the turbulent boundary layer canbe given implicitly for compressible flows

0.242√c f

(1 + γ − 1

2Ma2

∞

)− 12

= log(Rex c f ) − 1

2log

(1 + γ − 1

2Ma2

∞

). (42)

Results of the turbulent boundary layer at Ma∞ = 0.3 and ReL = 106 areshown in Figure 9 for the new and the Spalart–Allmaras model in comparison


with the log law distribution. Note that the beginning of the logarithmic region ispredicted at about y+ ≈ 30 with the new model, whereas the S A model predictsthis region earlier at y+ ≈ 10. The grid resolution study shows that the law ofthe wall is well predicted even on very coarse meshes for both turbulence models.The minimum grid spacing in the wall normal direction for the very coarse gridhas the normalized size �y+ ≈ 1 and a maximum stretching factor normal tothe wall of approximately 2.6. The good behavior of the turbulence models onrelatively coarse grids is one of the key factors for the success of turbulence modelssuch as the Spalart–Allmaras model. The skin-friction distributions c f , which aredepicted in logarithmic scale in Figure 10, evidence the transitional behavior of theturbulence model along the flat plate. Both models predict a natural transition froma laminar to a turbulent boundary layer. The Spalart–Allmaras model5 has a muchlonger transition period than the new model. The latter occurs at a Reynolds numberalmost one order of magnitude below the natural transition of the hydraulicly smoothflat plate Retransition = 3.5 × 105–106 [48]. Following the arguments of Spalart [52]the transition predicted by turbulence models lacking a detailed stability analysisare not generally valid and should not be trusted. However, it can to a certain extentbe advantageous to have a more upstream located transition to allow a comparisonwith fully turbulent flows.

Moreover, note that the turbulent skin-friction coefficient c f predicted by thenew turbulence model is slightly smaller than the theoretical distribution. This isdue to the chosen viscous damping function fv1. Nevertheless, the function fv1 isa good compromise since it does not depend on any wall distance d and still givesacceptable distributions even on the very coarse grid compared to the Spalart–Allmaras model as shown in Figure 10.

4.4. TURBULENT FLOW WITH PRESSURE GRADIENT

In typical boundary layer flows pressure gradients occur. There exist numer-ous test cases to measure the predictive capabilities of turbulence models forsuch flows. Figures 11–16 show the velocity and the skin-friction coefficientdistributions predicted by the k-ω, the Spalart–Allmaras, and the new turbu-lence model compared to measurements. The boundary layer was resolved nu-merically by 200 grid points and y+

min ≈ 1. The test cases were defined at theAFSOR–IFP–Stanford conference on the computation of Turbulent Boundary Lay-ers [19, 62]. Table III gives an overview on the calculated flows and the relatedexperiments.

The figures prove the capability of the novel turbulence model to com-pete with the other models. The new model behaves either like the k-ω or theSpalart–Allmaras model. Although the skin-friction coefficient distributions issomewhat overpredicted for the adverse pressure gradient flows (Figures 13,15, 16) it is fair to conclude that it yields in all cases satisfactory or goodresults.


Figure 9. Comparison of numerical findings and the law of the wall of a flat plate at Ma∞ = 0.3and ReL = 106 at different grid resolutions Top: (129×65), Center: (65×33), Bottom: (33×17).


Figure 10. Comparison of numerical findings of a flat plate flow at Ma∞ = 0.3 and ReL =106 and theoretical laminar and turbulent (Equation (42)) c f distributions at different gridresolutions Top: (129 × 65), Center: (65 × 33), Bottom: (33 × 17).


Figure 11. Computed and measured velocity (left) and skin-friction (right) for boundary layerflow at a favorable pressure gradient (Flow 1300 [24, 62]).

Figure 12. Computed and measured velocity (left) and skin-friction (right) for boundary layerflow at a favorable pressure gradient (Flow 6300 [62]).

4.5. RAE 2822 AIRFOIL

The transonic turbulent flow over the RAE 2822 airfoil, which was experimen-tally investigated in [7], represents a standard aerodynamic test case for turbulencemodels. The findings of the simulation of the flow at Ma∞ = 0.73, Rec = 6.5×106

and an angle of attack α = 2.79◦ using the Spalart–Allmaras and the new turbulencemodel are depicted in Figure 17. The distribution of the pressure coefficientdistribution cp of the new turbulence model shows better agreement with the exper-iments than the Spalart–Allmaras model especially in the shock region on the uppersurface and near the trailing edge on the lower surface. Note that the original paperof Spalart and Allmaras [52] shows the same deviation between the calculations us-ing the Spalart–Allmaras model and the measurements. For the c f distribution the


Figure 13. Computed and measured velocity (left) and skin-friction (right) for boundary layerflow at a weak adverse pressure gradient (Flow 1100 [24, 62]).

Figure 14. Computed and measured velocity (left) and skin-friction (right) for boundary layerflow at a weak adverse pressure gradient (Flow 2100 [49, 62]).

new model yields a slightly better agreement upstream of the shock region, whereasthe Spalart–Allmaras model shows better predictions downstream of the shock. Itcan be conjectured that the stronger shock predicted by the Spalart–Allmaras leadsto a smaller local Mach number after the shock. Consequently, the local velocitygradients normal to the wall and hence the skin-friction is smaller than that de-termined by the new turbulence model. In Figure 18 it can be seen by the almostidentical local Mach number contours near the airfoil for both turbulence modelsthat this result is very local and has hardly any impact on the overall flow field.

4.6. HIGH LIFT CONFIGURATION

The 2D flow of the multi-element airfoil still constitutes a challenging taskfor the computational fluid dynamics community since details like grid quality


Table III. Calculated boundary layer flows with pressuregradient.

Flow Description

Flow1300 Favorable pressure gradient [24]

Flow6300 Favorable pressure gradient [62]

Flow1100 Weak adverse pressure gradient [24]

Flow2100 Weak adverse pressure gradient [49]

Flow0141 Increasingly adverse pressure gradient [47]

Flow1200 Strong adverse pressure gradient [24]

Figure 15. Computed and measured velocity (left) and skin-friction (right) for boundary layerflow at an increasing adverse pressure gradient (Flow 0141 [47, 62]).

Figure 16. Computed and measured velocity (left) and skin-friction (right) for boundary layerflow at a strong adverse pressure gradient (Flow 1200 [24, 62]).


Figure 17. Computed and measured pressure coefficient cp distribution (left) and skin-frictioncoefficient c f distribution (right) of the RAE 2822 airfoil at Ma∞ = 0.73, Rec = 6.5 × 106,and α = 2.79◦.

Figure 18. Computed Mach number contours of the RAE 2822 airfoil at Ma∞ = 0.73, Rec =6.5 × 106, and α = 2.79◦.

and turbulence modeling play a significant role in the correct simulation of theflow. The airfoil geometry is the BAC 3 − 11 in high lift configuration (L1/T 2)as described in [29]. The geometry and the multiblock grid used for the numericalsimulation is shown in Figure 19.

Figure 20 depicts the pressure coefficient distribution on the surface comparedto experimental data from Moir [29] at Ma∞ = 0.198, Rec = 3.52 × 106,and α = 4◦. Except in the recirculation area on the lower slat surface a goodagreement is achieved with the new model showing a somewhat closer correspon-dence with the experimental data in this region than the S A model. A separatedunsteady flow in this area cannot be correctly captured by the simulation thatuses local time stepping. The Mach number distributions in Figure 21 show thewell-resolved wake of the slat and the main wing although no special grid re-finement study was preformed. Furthermore, the distributions emphasize the al-most identical simulation results of both turbulence models for the overall flowfield.


Figure 19. Multiblock grid of the three-element airfoil, minimum normal grid size �y = 10−6,19 blocks, 42000 nodes.

Figure 20. Computed and measured pressure coefficient cp distributions over the BAC 3-11airfoil in high lift configuration at Ma∞ = 0.198, Rec = 3.52 × 106, and α = 4◦ determinedby the new model and the S A model.

4.7. SENSITIVITY ANALYSIS OF THE MODEL

An analysis of the sensitivity of the results to the freestream turbulence and to gridresolution was done for the new model following the methodology in [3]. The valueof the dimensionless freestream eddy viscosity can be defined as νt/U x = 10n ,where the exponent n is varied within the range of −13 ≤ n ≤ 0. Furthermore,the percentage of deviation of the spreading rate value S compared to that obtainedon the finest grid resolution 100(S − S1000)/S1000 can be evaluated for differentresolutions ranging from 100–1000 grid points. The investigations are shown ex-emplarly for the mixing layer flow in Figure 22. As can be easily seen the newmodel shows a consistent behavior almost independent of the freestream value andthe grid resolution when compared to the k-ω or the S A models. Investigations ofother types of flows confirm the same behavoir of the new model.


Figure 21. Computed Mach number contours for the BAC 3-11 airfoil in high lift configurationat Ma∞ = 0.198, Rec = 3.52 × 106, and α = 4◦.

4.8. NUMERICAL ROBUSTNESS AND EFFICIENCY

The numerical implementation of the proposed model is very similar to that of theS A model. From a computational point of view the new model possesses slightlymore complicated terms that require additional computational time. However, thenew model doesn’t require the wall distance. The startup distance calculation istherefore not necessary, which can be expensive in the case of complicated 3Dgeometries [11]. Using the same stability CFL limit and achieving almost similarconvergence rates during the calculation the difference in the CPU time betweensimulations using the S A and the proposed model was within 4%. Comparisons withthe k-ω model confirm that this two equation model has more restrictive stabilitylimitations and considerably higher demands on the computational resources andgrid resolution especially near the wall.


Figure 22. Sensitivity of spreading rate to freestream turbulence (left) and grid resolution(right) for mixing layer flow.

5. Conclusion

A new one-equation model based on the k-ω model is derived. The turbulence modelis validated via numerous analytic and experimentally investigated flows to ensurethe generality of the numerical results. The findings show the novel one-equationturbulence model to predict a wide range of flows especially jets and vortical flowsmore accurately than the Spalart–Allmaras model while retaining the same qualityof results for near-wall flows and to be more efficient than the k-ω two-equationmodel.

Acknowledgments

This work has been performed within the collaborative research program 401 at theAachen University funded by the Deutsche Forschungsgemeinschaft (DFG) undergrant SFB401/TP A5.

Notes

1. Here we chose to eliminate k from the equations. Other choices are also possible but not describedhere.

2. Note the relation Cµ = β∗c = β2

b = 0.09 with Cµ being a closure coefficient of the k-ε model.3. Note that this behavior is similar to the multilayer structure of a turbulent boundary layer.4. According to [54] the given definition of the rotation terms in this paper may be incorrect.5. Note that the tripping function proposed in the original paper [52] is not used here.

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