The current state-of-the-art in accounting for mean property variations in compressible turbulent wall-bounded flows is the Van Driest transformation, which is inaccurate for non-adiabatic walls. An alternative transformation is derived, based on arguments about log-layer scaling and near-wall momentum conservation. The transformation is tested on supersonic turbulent channel flows and boundary layers, and it is found to produce an excellent collapse of the mean velocity profile at different Reynolds numbers, Mach numbers, and rates of wall heat transfer. In addition, the proposed transformation mathematically derives the semi-local scaling of the wall-normal coordinate and unifies the scaling of the velocity, the Reynolds stresses, and the wall-normal coordinate.

## I. INTRODUCTION

A powerful idea in the field of wall-bounded turbulent flows is the rough universality of the inner layer profiles when expressed in an inner scaling. Here, the most important aspect is the existence of a “law-of-the-wall” for the mean velocity profile. This idea was gradually discovered and derived by Prandtl and von Kármán^{2–4} in the context of incompressible, constant-property flows. In this context, the law-of-the-wall is

where the function *f* is an approximately universal function, *u* is the mean velocity, *y* is the wall-normal coordinate, *ρ* is the density, *μ* is the viscosity, and *u*_{τ} is the friction velocity. To within small experimental uncertainties or Reynolds number effects (cf. Ref. 5), the validity of the law-of-the-wall in the inner layer (say, up to 10%–20% of the boundary layer thickness *δ*) is well established.

In compressible flows, viscous heating causes non-uniform mean density and viscosity, which results in a mean velocity profile that no longer satisfies the law-of-the-wall. Much research has sought to extend the law-of-the-wall to this regime,^{6–15} and the most successful of these attempts have tackled the problem by seeking to transform the compressible velocity profile into an equivalent “incompressible” profile that satisfies the law-of-the-wall. This approach is natural, generalizing the simple scaling of *u* and *y* in Equation (1) by more general transformations.

Essentially, all such transformations rely implicitly on the observation that the dominant effect of compressibility is mean property variation, most importantly of the density and viscosity, with acoustic effects often having minor effects on the mean velocity profile.^{16,17}

The current state-of-the-art in describing a compressible mean velocity profile is to assume that Equation (1) is valid provided the scaled velocity *u*^{+} = *u*/*u*_{τ} is replaced by the so-called “Van Driest transformed” velocity

where *u*^{+} = *u*/*u*_{τ}. The transformation adjusts the velocity gradient by the factor $\rho \u0304/\rho w$, which is derived by dimensional reasoning in the log-layer. The transformed velocity is then used in Equation (1) with the coordinate $y+=y\rho w\tau w/\rho w/\mu w$.

Who exactly developed what is now called the Van Driest transformation is a matter of debate. Several pioneering researchers, most notably Van Driest,^{10} developed roughly similar trigonometric transformations during the early 1950s. These forms all derive from the log-law but differ in their velocity-temperature couplings. The integral form came later. By the early 1960s, several textbooks (for example, Dorrance^{18} and Kutateladze and Leont’ev^{19}) contained this integral equation but interpreted it as part of a skin friction formula. The modern interpretation of this integral as a velocity transformation came from Danberg,^{15} who realized that it represented the ultimate generalization of the velocity transformation work of Van Driest^{10} and others.

### A. Accuracy of the Van Driest transformation for adiabatic and cooled walls

The status of the Van Driest transformation as the current state-of-the-art stems from its accuracy in boundary layers over adiabatic walls. An example of its ability to restore approximate universality—that is, that the velocity transformed according to Equation (2) satisfies the law-of-the-wall—is shown in Figure 1. The collapse between the different cases in the inner layer (in the viscous sublayer, the buffer layer, and the log-layer) is clear. More broadly, the validity and accuracy of the Van Driest transformation for adiabatic walls up to Mach 20 has been confirmed in many experiments^{23–25} and direct numerical simulations (DNS).^{20,26–31} The collapse in the log-layer specifically means that the Van Driest transformed velocity follows an approximately universal log-law, with von Kármán constant *κ* and log-law intercept *C* that do not depend on the Mach number.^{15}

The accuracy of the Van Driest transformation deteriorates for increasingly non-adiabatic walls (that is, walls with heat leaving or entering the fluid). Most practical applications have cooled walls, where the wall heat transfer rate *q _{w}* < 0. The relevant dimensionless measure of the wall heat transfer rate for wall-turbulence is

^{13}

which measures temperature variation in the viscous sublayer. An example of the effects of wall-cooling on the Van Driest transformed velocity is shown in Figure 2 (which plots data from channel DNS). As the cooling increases (increasingly negative *B _{q}*), three distinct changes distort the figure: (1) an upwards shift in the log-law intercept

*C*; (2) a drop in the slope

*S*in the viscous sublayer; and (3) an outwards shift in the wall-normal coordinate signifying a thickened buffer layer. The changes are minor at

*B*= − 0.053 but quite severe at

_{q}*B*= − 0.131.

_{q}To quantify these changes, the extracted log-law intercept is defined as

where *U*^{+} is the transformed velocity and the integration bounds are chosen to be in the log-layer or slightly above. The slope in the viscous sublayer is defined as the mean slope between the wall and *Y*^{+} = 4 (that is, the value of *U*^{+} at *Y*^{+} = 4 divided by 4),

These quantities are shown in Figure 3 for the present channel DNS data. The changes in both quantities are essentially linear functions of −*B _{q}*.

The inability of the Van Driest transformation to collapse data for cooled walls has been known since at least the 1960s. Danberg^{15} noticed the near perfect collapse of adiabatic boundary layers using the Van Driest transformation, but remarked that “*C* increases quite rapidly with heat transfer into the wall,” though his skin friction estimation exaggerated the effect. However, later DNS of cooled wall turbulent flows^{27,33,34} confirmed this trend, with exception of Duan *et al.*^{35} who did not observe much of an upwards shift and concluded that “the additive constant is relatively insensitive to wall-temperature conditions.”

The drop in the viscous sublayer slope *S* was noticed in the DNS studies mentioned above, and also in some experiments with sufficient near-wall resolution.^{21} The Van Driest transformation is derived to produce the correct slope in the log-layer, and similarly a viscous sublayer transformation can be derived,^{36}

to ensure that the transformed $UVS+\u2248y+$ in the viscous sublayer. The challenge is to find a transformation that produces the correct “universal” slopes in both the viscous sublayer and the log-layer, and also that integrates to the correct log-layer intercept.

### B. Current paradigm for compressible wall turbulence

The discussion above was limited to the mean velocity profile and how it is transformed to (hopefully) satisfy the law-of-the-wall. This idea is one part of a broader set of ideas that make up the current paradigm of scalings for compressible wall turbulence. The current paradigm is

*Y*^{+} is the transformed wall-normal coordinate, *U*^{+} is the transformed mean velocity, *R*^{+} is the transformed Reynolds stress (this result was also prefigured by Rotta^{41}), and Re_{δ2} is a Reynolds number that controls for flow similarity.^{40}

Several generations developed the current paradigm carefully, adding new pieces and replacing old pieces when needed. Still, the current paradigm cannot unite all the ideas contained within it. For example, while using two different coordinates works, it is hard to believe that the turbulence lives in one domain and the mean flow in another. Nonetheless, the two coordinate paradigm has worked and continues to work in practice.

### C. Objectives and outline

The objective of the present work is to determine how to transform a mean velocity profile such that it satisfies the law-of-the-wall (Equation (1)) regardless of the Reynolds number, the Mach number, or the wall heat transfer rate. The proposed transformation is derived in Section II and assessed in Section III. In the end, a new “paradigm” emerges with a single transformed coordinate which collapses both mean velocity and Reynolds stresses.

## II. DERIVATION OF PROPOSED TRANSFORMATION

This paper derives the proposed transformation in three parts. The first part derives a condition from the log-law, the second part derives a stress balance condition, and the third part combines these two conditions to obtain the full transformation.

The objective seeks to distinguish and relate two different states: an incompressible state with constant properties (the transformed state), and a compressible state with variable properties (the untransformed state). The “raw” compressible state has mean velocity *u*, density $\rho \u0304$, viscosity $\mu \u0304$, Reynolds stress $rij=ui\u2032uj\u2032\u02dc$, and wall-normal coordinate *y*. The transformed state is defined (by convention) to have constant values of density and viscosity equal to the wall-values of the compressible state, so it has density *ρ _{w}* and viscosity

*μ*. By definition, the two states share the same wall shear stress

_{w}*τ*.

_{w}For the remaining quantities, the raw compressible state is referred to by lowercase letters (velocity *u*, wall-normal coordinate *y*, and Reynolds stress $rij=ui\u2032uj\u2032\u02dc$), while the transformed state (still in dimensional form) is referred to by uppercase letters or a “*” superscript (transformed velocity *U*, wall-normal coordinate *Y*, and Reynolds stress *R _{ij}*).

The “+” superscript denotes a dimensionless quantity, scaled by the corresponding scale value at the wall. Since both states share the same wall shear stress and the same density and viscosity at the wall, both states share the same friction velocity *u*_{τ} and viscous length scale ℓ_{ν}, which are

From these two scales, the velocity scale is *u*_{τ}, the coordinate scale is ℓ_{ν}, and the Reynolds stress scale is $u\tau 2$.

In all, transforming from the raw quantities to the approximately universal quantities implies transforming *y* → *Y*^{+} and *u* → *U*^{+}.

Most research has approached this objective differentially. A differential transformation is more general than a simple scaling, like the classical law-of-the-wall scaling. But unlike a simple scaling, a differential transformation follows the chain rule, which reveals how different parts of the transformation behave,

So the transformed velocity gradient *dU*/*dY* is a function of the velocity transformation kernel *dU*/*du*, the coordinate transformation kernel *dY*/*dy*, and the untransformed velocity gradient *du*/*dy*. The transformed coordinate or velocity is then obtained by integrating the transformation kernels,

Here, the wall is assumed to have *u* = 0 and be at *y* = 0. The definition of the dimensionless units (Equations (7) and (8)) gives the transformation kernels used here the property that

so brevity motivates using only the dimensional form without plus-units.

This framework has been used in prior studies, and it contains most existing transformations. For example, the Van Driest transformation results from taking *dY*/*dy* = 1 and $dU/du=(\rho \u0304/\rho w)1/2$, while the viscous sublayer transformation results from taking *dY*/*dy* = 1 and $dU/du=(\mu \u0304/\mu w)$. The transformation by Brun *et al.*^{42} can also be written in this format. In addition, the laminar coordinate transformation of Cope and Hartree^{43} is $dY/dy=\mu w/\mu \u0304$, and the laminar coordinate transformation of Howarth^{44} is $dY/dy=\rho \u0304/\rho w$.

### A. Derivation of log-law condition

A transformation condition derived from the log-law has been studied for decades. Van Driest^{10} and Danberg^{15} considered a similar condition, which they used to derive the Van Driest transformation. More recently, Brun *et al.*^{42} considered a more general form of this condition that includes the possibility of a coordinate transformation. This section generalizes all of this previous work without presupposing a particular coordinate transformation.

Consider the velocity gradient in the log-layer, in which viscous effects are, by definition, unimportant. Following the dimensional reasoning of Bradshaw,^{45} the relevant variables are the shear stress at the wall *τ _{w}*, the local density $\rho \u0304$, and the distance from the wall

*y*, as per Townsend’s attached eddy hypothesis (and assuming that turbulent length scales are unaffected by property variations). Dimensional analysis then yields the velocity scale $\tau w/\rho \u0304$ and length scale

*y*. In the form of a velocity gradient, these yield

Multiply the compressible gradient by *Y*/*Y* and (*ρ _{w}*/

*ρ*)

_{w}^{1/2}, then rearrange and group terms. The incompressible log-law velocity gradient appears on the right hand side. Upon rearrangement and simplification, the incompressible velocity gradient comes to

or, in terms of the velocity transformation kernel,

What distinguishes the present work from prior works is that the coordinate transformation kernel *dY*/*dy* is left as an unknown to solve for—in no way is it presupposed or provided. For example, the work leading up to the Van Driest transformation implicitly assumed *Y* = *y*, while the more recent work of Brun *et al.*^{42} explicitly assumed the laminar coordinate transformation of Cope and Hartree.^{43} In contrast, Sec. II B seeks to apply a second condition to derive the correct coordinate without presupposing its definition.

### B. Derivation of stress balance condition

Earlier, the chain rule (Equation (9)) revealed that all similar differential transformations—including both the Van Driest and viscous sublayer transformations—operate directly on velocity gradients, and only indirectly operate on the velocities themselves (through integration).

This observation motivates an additional condition involving the velocity gradients themselves. In effect, the Van Driest transformation adjusts the velocity gradients without worrying about the underlying physical mechanisms that determine their values. In other words, the Van Driest transformation obtains the correct slope while violating the stress balance (momentum conservation), since the velocity gradients determine the viscous stresses. Therefore, any realistic transformation must preserve the stress balance between the untransformed and transformed states.

Given the connection between the gradients and the viscous stresses, it seems natural to consider the balance of shear stresses in the inner layer for a second condition. The transformed state should satisfy the incompressible, constant property equations of motion to remain physically relevant. Simply stated, it should satisfy momentum conservation, which in the inner layer of nearly parallel shear flows at reasonable turbulence Mach numbers boils down to the stress balance equation,^{46,47}

Here, a tilde denotes a mass-averaged quantity and a prime denotes a fluctuation from the mass-averaged quantity (see Refs. 46 and 47). Still, the incompressible state and compressible state must be linked somehow. By construction, the wall shear stress *τ _{w}* is identical in both the raw and transformed states, and thus the sum of viscous and Reynolds stresses in both the raw and transformed states must be equal,

This equation is quite powerful. In fact, it embodies two seemingly disconnected ideas: Morkovin’s scaling^{38} for the Reynolds stresses and the viscous sublayer transformation. And it also derives the stress balance condition used in the proposed transformation.

Morkovin’s scaling derives from assuming that the turbulent shear stresses are roughly equal in magnitude in the flow’s most turbulent region. Here, the viscous stresses are negligible and Equation (15) simplifies to

When non-dimensionalized by dividing by $u\tau 2$ and re-arranged, this equation becomes Morkovin’s scaling,^{38}

This scaling is well-known to properly scale the Reynolds stresses. The derivation continues by assuming that Morkovin’s scaling applies everywhere, even in the viscous sublayer or buffer layer. Moreover, unlike previous research,^{42} the velocity transformation and the Reynolds stress scaling are not assumed to be related at all. Assuming that Morkovin’s scaling holds everywhere—that $\rho \u0304ruv=\mu wRuv$—simplifies the final stress balance condition to an adjustment of the velocity gradients only,

What this hypothesis means is that provided Morkovin’s scaling holds, the viscous stresses must also remain equal to maintain momentum conservation in the transformed state. The incompressible velocity gradient comes to

and the velocity transformation kernel becomes

The important part here is that the total stress balance (including both the viscous and Reynolds stresses) must hold for the entire inner layer, and not just for the viscous sublayer. Specifically, while Equation (20) becomes the viscous sublayer transformation for *dY*/*dy* = 1, that transformation was derived by assuming zero Reynolds shear stress near the wall. In the present reasoning, the Reynolds shear stress was never assumed to be zero; it was merely assumed to be identical between the compressible and transformed states. Similarly, these equations do not imply that viscous effects are important in the log-layer, but merely that, if the total stress is the same in two states, and if the Reynolds shear stress is the same in those two states (again, if Morkovin’s scaling holds), then the viscous stresses must also be equal in those two states.

### C. Final steps to derive proposed transformation

The log-law condition and the stress-balance condition form two equations with two unknowns (the transformation kernels), so they are easily solved. Setting them equal yields

which upon rearrangement reveals the transformed and dimensional coordinate *Y*,

The universal coordinate (transformed and dimensionless) then comes to

This equation is the semi-local scaling (often termed *y*^{∗}), which the current paradigm uses to collapse Reynolds stresses. It had been used since the 1950s,^{41,48} but the most important observation about it came from Huang *et al.*^{37} and its sibling paper of Coleman *et al.*,^{33} which found (by observation) that this coordinate led to an improved collapse of the velocity fluctuations (however, they still used the standard *y*^{+} and the Van Driest transformation for the velocity).

In fact, this coordinate is a direct consequence of assuming that the transformed log-law obeys a transformed stress balance (what the two conditions jointly imply). For that reason, the present derivation suggests that the semi-local scaling is the correct coordinate for all quantities. The present derivation involved the coordinate, the velocities, and the Reynolds stresses in a unified manner, so this coordinate—once thought to work for the fluctuations only—should work for both velocities and Reynolds stresses, provided the corresponding velocity transformation is found.

Now that the transformed coordinate *Y* has been derived, the velocity transformation kernel *dU*/*du* can be found in multiple ways. For example, differentiation of Equation (22) yields the coordinate transformation kernel,

The proposed velocity transformation is simply the integral of this equation. The (nearly) complete proposed transformation then is this transformed velocity, the transformed dimensionless coordinate *Y*^{+} from Equation (23), and the Morkovin’s scaling for the Reynolds stresses from Equation (17).

The proposed transformation gives rise to a new Reynolds number. In incompressible flows, the friction Reynolds number Re_{τ} is the ratio of the boundary layer thickness *δ* (or channel half-height or pipe radius) to the viscous length scale. As such, it directly measures the viscous effects on the largest scales. The largest scales in a compressible flow should be equally affected by viscosity if the friction Reynolds number of the transformed state is the same; therefore, the transformed friction Reynolds number is

So, in summary, the complete transformation in closed-form is

## III. ASSESSMENT OF PROPOSED TRANSFORMATION

Now that the proposed transformation has been derived, the main question left concerns how well this transformation performs. This section answers that question using both numerical data (from the present channel DNS and from the boundary layer DNS of Duan and Martán^{49}) and experimental data (from data in Refs. 23 and 25), but it also seeks to answer what limits how well the transformation works (its robustness). Finally, this section discusses the transformed Reynolds number $Re\tau \u2217$ and its use as a flow similarity parameter before concluding.

### A. DNS of compressible channels flows

Compressible turbulent channel flows offer a computationally affordable platform for studying the effects of wall-cooling on wall-bounded turbulence. This kind of flow was studied in the important prior works of Coleman *et al.*,^{33} Huang *et al.*,^{37} and Foysi *et al.*^{34} The flow between two infinite parallel plates is driven by a body force which performs work on the flow. This injected energy can only leave the domain through wall heat transfer, and thus compressible channels necessarily have cooled walls (at statistical steady state). The energy balance causes the cooling parameter *B _{q}* defined in Equation (3) to be $Bq=\u2212(\gamma \u22121)Mab2/Ub+$, that is, there is a direct relationship between the bulk Mach number Ma

_{b}=

*u*

_{avg}(

*γRT*)

_{w}^{−1/2}and the cooling-rate. The 9 cases in the present study are listed in Table I. The numerical details of the simulations (code, grid-spacing, domain size, etc.) are described in the Appendix.

Case name . | Ma_{b}
. | Re_{h}
. | Re_{τ}
. | $Re\tau \u2217$ . | −B
. _{q} | C_{VD}
. | C_{proposed}
. | C_{ref}
. | S_{VD}
. | S_{VS}
. | S_{proposed}
. |
---|---|---|---|---|---|---|---|---|---|---|---|

M0.7R400 | 0.7 | 7 500 | 437 | 396 | 0.011 | 5.592 | 5.472 | 5.333 | 0.963 | 0.978 | 0.978 |

M0.7R600 | 0.7 | 11 750 | 652 | 591 | 0.010 | 5.499 | 5.384 | 5.399 | 0.963 | 0.978 | 0.978 |

M1.7R200 | 1.7 | 4 500 | 322 | 197 | 0.057 | 6.716 | 6.017 | 5.736 | 0.902 | 0.981 | 0.977 |

M1.7R400 | 1.7 | 10 000 | 663 | 406 | 0.053 | 6.040 | 5.427 | 5.336 | 0.910 | 0.984 | 0.982 |

M1.7R600 | 1.7 | 15 500 | 972 | 596 | 0.050 | 6.080 | 5.461 | 5.399 | 0.913 | 0.982 | 0.978 |

M3.0R200 | 3.0 | 7 500 | 650 | 208 | 0.131 | 7.503 | 5.913 | 5.711 | 0.824 | 0.982 | 0.976 |

M3.0R400 | 3.0 | 15 000 | 1232 | 396 | 0.123 | 6.937 | 5.429 | 5.333 | 0.832 | 0.982 | 0.976 |

M3.0R600 | 3.0 | 24 000 | 1876 | 601 | 0.116 | 6.894 | 5.406 | 5.399 | 0.839 | 0.983 | 0.977 |

M4.0R200 | 4.0 | 10 000 | 1017 | 203 | 0.189 | 8.020 | 5.883 | 5.723 | 0.780 | 0.992 | 0.984 |

Case name . | Ma_{b}
. | Re_{h}
. | Re_{τ}
. | $Re\tau \u2217$ . | −B
. _{q} | C_{VD}
. | C_{proposed}
. | C_{ref}
. | S_{VD}
. | S_{VS}
. | S_{proposed}
. |
---|---|---|---|---|---|---|---|---|---|---|---|

M0.7R400 | 0.7 | 7 500 | 437 | 396 | 0.011 | 5.592 | 5.472 | 5.333 | 0.963 | 0.978 | 0.978 |

M0.7R600 | 0.7 | 11 750 | 652 | 591 | 0.010 | 5.499 | 5.384 | 5.399 | 0.963 | 0.978 | 0.978 |

M1.7R200 | 1.7 | 4 500 | 322 | 197 | 0.057 | 6.716 | 6.017 | 5.736 | 0.902 | 0.981 | 0.977 |

M1.7R400 | 1.7 | 10 000 | 663 | 406 | 0.053 | 6.040 | 5.427 | 5.336 | 0.910 | 0.984 | 0.982 |

M1.7R600 | 1.7 | 15 500 | 972 | 596 | 0.050 | 6.080 | 5.461 | 5.399 | 0.913 | 0.982 | 0.978 |

M3.0R200 | 3.0 | 7 500 | 650 | 208 | 0.131 | 7.503 | 5.913 | 5.711 | 0.824 | 0.982 | 0.976 |

M3.0R400 | 3.0 | 15 000 | 1232 | 396 | 0.123 | 6.937 | 5.429 | 5.333 | 0.832 | 0.982 | 0.976 |

M3.0R600 | 3.0 | 24 000 | 1876 | 601 | 0.116 | 6.894 | 5.406 | 5.399 | 0.839 | 0.983 | 0.977 |

M4.0R200 | 4.0 | 10 000 | 1017 | 203 | 0.189 | 8.020 | 5.883 | 5.723 | 0.780 | 0.992 | 0.984 |

The accuracy of the Van Driest transformation and the proposed one is compared in Figure 4 for the highest Reynolds number. The profiles transformed by the proposed transformation are almost indistinguishable from the incompressible reference, implying that the proposed transformation recovers an approximately universal law-of-the-wall.

The most strongly cooled cases at each Reynolds number are shown in Figure 5, which plots both the mean velocity and the shear stresses. The shear stresses collapse almost as well as the mean velocity, which in hindsight is unsurprising: the mean velocity profile in the inner layer is a direct consequence of the variation and balance between the viscous and turbulent shear stresses. The inclusion of the stress-balance condition (Equation (15)) in the derivation essentially enforces that

in the transformed state.

### B. Robustness of the transformation

Now that the accuracy of the proposed transformation has been assessed, it is necessary to assess its numerical robustness. The difference between accuracy and robustness is subtle. A transformation is accurate if it correctly transforms the data, and a specific implementation is robust if it correctly transforms data even when the data set is sparse or poor.

This work considers two different ways of implementing the velocity transformation integral. The first method (dubbed the “long” method here) uses the velocity transformation kernel *dU*/*du* in the form given in Equation (25). A second method (dubbed the “short” method), which requires the evaluation of fewer property gradients, first computes the transformed coordinate *Y* from Equation (22), then differentiates this coordinate numerically to form *dY*/*dy* = *dY*^{+}/*dy*^{+}, and then uses the velocity integration kernel in the form of Equation (20) to evaluate the integral

using some quadrature method.

The most plausible situation in practice (especially in experiments) is to have poor resolution near the wall. To mimic this situation, two data sets—an adiabatic one and a cooled one—are truncated by removing data points within a distance Δ*Y*^{+}|_{w} from the wall before applying the transformation machinery.

The metric used to assess the robustness is the log-law intercept *C* in the transformed state. *C* is a reasonable metric partly because it directly relates to the skin friction (and therefore has direct physical importance), and partly because the profiles will be well-resolved in the log-layer (and therefore the slope there should be accurate and insensitive to incomplete data or other errors). The variation of *C* with increasingly incomplete data (that is, with increasing values of Δ*Y*^{+}|_{w}) is shown in Figure 6, which prompts several important and interesting observations.

The robustness barely changes with Δ*Y*^{+}|_{w} for the adiabatic wall, particularly for the Van Driest transformation but also with the two different numerical implementations of the proposed one. The small variation is likely due to the fact that the property-gradients should approach 0 at an adiabatic wall, and thus the small variations in the viscous sublayer and buffer layer do not cause the transformation kernels to deviate from unity by much. This observation also explains why the Van Driest transformation works for adiabatic walls: while its transformation kernel is incorrect below the log-layer, the property variations there are sufficiently small as to avoid affecting the solution by much.

For the cooled wall, the Van Driest transformation is again rather robust, but, of course, inaccurate. Interestingly, the two proposed implementation methods differ substantially here, with the “short” form behaving much more robustly. Depending on the tolerance of error, the “long” form remains accurate for Δ*Y*^{+}|_{w} ≲ 5 while the “short” form remains accurate for Δ*Y*^{+}|_{w} ≲ 10, for this specific cooling-rate. For lower cooling-rates, the robustness likely approaches that of the adiabatic case.

Ultimately, Figure 6 bears a major conclusion and warning: even if an experiment measures *τ _{w}* or

*c*correctly, it may transform data incorrectly if the first data point is too far from the wall (with the critical value depending on the cooling-rate). This requirement makes sense, since stronger cooling implies more rapid variation and thus more stringent resolution requirements. Unfortunately, this requirement does set a high standard for any future experiments seeking to validate the proposed transformation.

_{f}### C. Validation of the proposed transformation in supersonic boundary layers

So far, the proposed transformation has been tested on supersonic channel flows, over a range of Reynolds numbers and Mach numbers. However, these cases do not allow for the Mach number and wall-cooling rate to be decoupled, so to investigate this effect, three boundary layer experiments and one boundary layer DNS are used to validate the transformation for adiabatic-wall and cooled-wall boundary layers.

The use of experimental data to validate the proposed transformation is made difficult by the resolution requirements discussed previously. However, data compiled in the work of Fernholz and Finley^{23} and Fernholz *et al.*^{25} do demonstrate the proposed transformation’s validity despite this requirement.

Table II details the three experimental cases and one numerical case used here: one nearly adiabatic case, two lightly cooled cases, and a strongly cooled DNS case from Duan and Martán.^{49} The mean velocity profiles are shown in Figure 7. All 3 experimental data sets include points below *Y*^{+} ≈ 10, which in accordance with the prior section ensures small numerical errors when evaluating the transformation integral (specifically since these cases at best mildly cooled). The DNS case’s wall resolution is Δ*y*^{+}|_{w} = 0.17, which is more than sufficient to validate the transformation.

Catalog number . | Ma_{e}
. | −B
. _{q} | Re_{τ}
. | $Re\tau \u2217$ . | Re_{δ2}
. | Re_{θ}
. | c ⋅ 10_{f}^{4}
. | S_{VD}
. | S_{proposed}
. | Source . |
---|---|---|---|---|---|---|---|---|---|---|

72020205 | 4.823 | 0.005 | 1549 | 13 925 | 6 962 | 28 764 | 7.392 | 1.039 | 1.026 | 21 |

72021501 | 4.929 | 0.069 | 9676 | 13 380 | 21 288 | 25 494 | 10.800 | 0.687 | 0.895 | 21 |

7702S0301 | 3.028 | 0.042 | 992 | 1 812 | 2 479 | 3 656 | 23.300 | … | … | 50 |

LowH_M3 | 3.400 | 0.200 | 938 | 409 | 1 554 | 958 | … | 0.776 | 0.948 | 49 |

Catalog number . | Ma_{e}
. | −B
. _{q} | Re_{τ}
. | $Re\tau \u2217$ . | Re_{δ2}
. | Re_{θ}
. | c ⋅ 10_{f}^{4}
. | S_{VD}
. | S_{proposed}
. | Source . |
---|---|---|---|---|---|---|---|---|---|---|

72020205 | 4.823 | 0.005 | 1549 | 13 925 | 6 962 | 28 764 | 7.392 | 1.039 | 1.026 | 21 |

72021501 | 4.929 | 0.069 | 9676 | 13 380 | 21 288 | 25 494 | 10.800 | 0.687 | 0.895 | 21 |

7702S0301 | 3.028 | 0.042 | 992 | 1 812 | 2 479 | 3 656 | 23.300 | … | … | 50 |

LowH_M3 | 3.400 | 0.200 | 938 | 409 | 1 554 | 958 | … | 0.776 | 0.948 | 49 |

The skin friction value for case 7702S0301 was taken from Fernholz *et al.*^{25} The skin friction value for 72020205 came from calculating the gradient at the second point in the data set (which gave the best fit to *U*^{+} = *y*^{+}). Given the high resolution of the data in the viscous sublayer, this approximation is reasonable. The skin friction value for 72021501 is the corrected value given in the work of Voisinet^{51} by way of Fernholz and Finley.^{24} Probe effects minutely distort the velocities in the viscous sublayer in 7702S0301, but the log-law values are intact and reasonable.

In all 3 experimental cases, the proposed transformation works as well as the Van Driest transformation in both the adiabatic and cooled situations. In general, both transformations match point by point up the incompressible reference profile. In all 3 cases, the level of heat transfer at the wall is quite small in fact—the cooled cases are at most comparable to the Mach 1.7 channel DNS cases given previously—so the fact that both transformations agree is unsurprising.

The proposed transformation only performed better in a single metric in the 3 experimental cases, the viscous sublayer slope *S*. Table II omits values of *S* for data sets without points in the viscous sublayer. The proposed transformation works noticeably better than the Van Driest transformation in 72021501. Here, the slope in the viscous sublayer is too low, and the proposed transformation corrects it within a reasonable level of experimental error.

In addition to these 3 experiments, a low-enthalpy case (that is, a calorically perfect gas case) from the DNS study of Duan and Martín^{49} was considered. This case was computed as a temporally evolving Mach 3.4 boundary layer at Re_{δ2} = 1554.5 with a wall temperature ratio of *T _{w}*/

*T*= 0.17 (

_{aw}*B*≈ − 0.2). The mean velocity profiles, transformed using both the Van Driest and the proposed transformations, are shown in Figure 7(d). For comparison, incompressible results at two different Reynolds numbers are shown: a profile at Re

_{q}_{τ}= 445 from Jiménez

*et al.*

^{22}and a profile at Re

_{τ}= 1310 from Sillero

*et al.*

^{52}

The results here are consistent with the channel flow DNS. The Van Driest transformed profile’s viscous sublayer slope is far too low—it drops nearly 25% below its incompressible value—and its log-law intercept is too high, while the proposed transformation produces excellent agreement especially in the viscous sublayer and through the buffer layer. Interestingly, while the channel DNS cases led to a perfect agreement in the log-layer with the proposed transformation, for this boundary layer case the log-layer mismatches slightly; the reason for this is unclear. Nevertheless, the proposed transformation drastically improves the collapse with the incompressible results.

### D. $Re\tau \u2217$ as a characteristic Reynolds number

At the end of the derivation, it was pointed out that the proposed transformation naturally leads to a transformed friction Reynolds number $Re\tau \u2217$ (Equation (26)), which is just the value of the transformed coordinate *Y*^{+} at the boundary layer’s edge. This Reynolds number is not unfamiliar—Cebeci and Bradshaw^{53} discussed it, for example—and it appears to properly characterize these wall-bounded flows (for example, see the recent work of Patel *et al.*^{54}). Here, “properly characterize” means that a compressible flow at $Re\tau \u2217$ corresponds to the equivalent incompressible flow at Re_{τ}.

One specific metric for assessing how well $Re\tau \u2217$ characterizes the flow is the log-law intercept in the transformed state. For incompressible channel flows, there is a known low Reynolds number effect in *C*, specifically that it increases slightly for very low Re_{τ}. Figure 8 shows this increase using the incompressible channel flow DNS data of Moser *et al.*,^{32} Hoyas and Jiménez,^{55} and Lee and Moser.^{56} For the Van Driest transformation, the Reynolds number is the standard Re_{τ}, and for the proposed transformation, the Reynolds number is $Re\tau \u2217$. The proposed transformation correctly picks up this low Reynolds number effect, while the van Driest transformation does not. This observation means that the proposed transformation accounts for Reynolds number effects as well, provided the transformed profile is compared to an equivalent incompressible profile with the same $Re\tau \u2217$. These results suggest that $Re\tau \u2217$ for compressible flows is the equivalent to Re_{τ} for incompressible flows.

The most common Reynolds number used to characterize compressible boundary layers is Re_{δ2} = *ρ _{e}u_{e}δ*

_{2}/

*μ*. To see how Re

_{w}_{δ2}relates to $Re\tau \u2217$, note that $Re\tau \u2217=Ree(cf/2)1/2$ or $Re\tau \u2217=(\rho eue\delta /\mu e)(cf/2)1/2$, where $cf/2=\tau w/(\rho eue2)$. Manipulation of Re

_{e}into Re

_{δ2}yields

The friction factor *c _{f}* is generally approximated (on empirical grounds) as being a function of Re

_{δ2}, and thus the last two factors are approximately functions of Re

_{δ2}. The first two factors should be insensitive to the Reynolds number, at least at sufficient high Reynolds numbers. They are, instead, directly dependent on the Mach number and the wall temperature boundary condition. As such, the Reynolds numbers $Re\tau \u2217$ and Re

_{δ2}should directly correspond for fixed Mach number and fixed wall thermal boundary conditions.

## IV. CONCLUSIONS

The main objective of this paper has been to derive a new transformation that relates a compressible mean velocity profile at arbitrary Mach number and wall thermal conditions to an approximately universal profile in the inner layer. This objective was accomplished by considering transformations for both the velocity and wall-coordinate simultaneously, and by specifying two separate conditions that then determine the transformation kernels.

The idea of simultaneously transforming both velocity and coordinate is old. The classic law-of-the-wall “transforms” these quantities by simply dividing them by the relevant factors (*u*_{τ} and ℓ_{ν}), and only works when both are scaled simultaneously. The proposed transformation differs only in that it allows for more general transformations than a simple scaling. Therefore, in the history of similar work, transformations like this one are more related to the idea of an inner layer scaling than to the more specific idea of a log-layer, despite the transformation (and the classic Van Driest transformation) using log-layer arguments directly.

The proposed transformation also unifies the scaling (or transformation) of the mean velocity and the Reynolds stresses, specifically by using the same “universal” coordinate for both. What matters mostly, however, is that this coordinate *emerged* naturally from the derivation without being specified or contrived. In hindsight, using two differently scaled coordinates for different physical quantities in the current “paradigm” seems to violate the basic ideas of an inner scaling for the physics as a whole, not just for any one quantity like velocity or Reynolds stress.

The proposed transformation was tested favorably on supersonic channel flows and on boundary layer DNS and experiments from the literature. This assessment should be seen as a first step, with more comprehensive validation being required. Specifically, boundary layers at higher Reynolds numbers and stronger cooling-rates should be studied. In addition, incompressible heated boundary layers would allow for the effects of viscosity- and density-variations to be decoupled from each other. The recent work of Lee *et al.*^{57} and Patel *et al.*^{54} is examples of such studies. This study limited itself to the inner layer only, so the transformation’s behaviour in the outer layer should be studied as well. The recent work of Zhang *et al.*,^{58} who studied the outer layer over adiabatic walls, is an example of such a study.

## Acknowledgments

The authors would like to acknowledge the help of several individuals and organizations during the pursuit of this research: Muhammed Atak, for help running the M3.0R600 case; the Library of Congress, for providing access to difficult to find literature; Ben Trettel, for procuring documents during the literature search and reviewing early drafts of this paper; and UMD department of IT, for computing time on UMD’s Deepthought 2 (special thanks to Jim Zahniser). The second author has been supported by NSF Grant No. CBET-1453633 for parts of this work. This research originally appeared in a MSc thesis.^{1}

### APPENDIX: NUMERICAL DETAILS FOR THE CHANNEL DNS

The compressible channel flow simulations are computed using the *Hybrid* code, which uses a solution-adaptive finite-difference method to solve the compressible Navier-Stokes equations for a perfect gas on Cartesian, stretched grids. For these simulations, the ratio of specific heats *γ* = *c _{p}*/

*c*= 1.4, the viscosity is assumed to follow the power-law

_{v}*μ*=

*μ*(

_{w}*T*/

*T*)

_{w}^{3/4}, and the Prandtl number is taken as 0.7. The code identifies shocks and other discontinuities using a shock-sensor based on comparing dilatation and vorticity, similarly to that proposed by Ducros

*et al.*

^{59}Shocks are treated by a 5th-order WENO scheme with Roe flux-splitting, while broadband turbulence motions (which, in these channel flow problems, constitute the vast majority of the domain) are treated by a 6th-order central difference scheme applied to a split form of the convective terms.

^{59}The use of the split form drastically improves nonlinear numerical stability, and no dealiasing filter is used. The solution-adaptivity creates internal scheme-boundaries, the stability of which has been analyzed and proved.

^{60}Time advancement is done using a 4th-order accurate explicit Runge-Kutta scheme. The numerical method has been verified, validated, and used on several problems in prior studies.

^{61,62}

The channel flows are driven by a spatially uniform body force *f*, which is adjusted at each time step to maintain a constant bulk velocity *u _{b}*. The domain size is [ℓ

_{x}, ℓ

_{y}, ℓ

_{z}]/

*h*= [10, 2, 3], where

*h*is the half-height. This is more than sufficient for the present study which is focused on the inner layer.

Table III lists all 9 DNS cases used in this study. The cases were chosen to cover different cooling parameters *B _{q}* from essentially adiabatic (Ma = 0.7,

*B*≈ − 0.010) to strongly cooled (Ma = 3.0 and 4.0,

_{q}*B*≈ − 0.12 and −0.19). In terms of Reynolds number, the cases were targeted at having transformed Reynolds numbers $Re\tau \u2217$ close to 200, 400, and 600—values close to the classic incompressible DNS cases of Moser

_{q}*et al.*

^{32}

Case name . | Ma_{b}
. | Re_{h}
. | Re_{τ}
. | $Re\tau \u2217$ . | −B
. _{q} | T/_{c}T
. _{w} | n
. _{x} | n
. _{y} | n
. _{z} | Δx^{+}
. | $\Delta ymin+$ . | Δy_{max}/h
. | Δz^{+}
. |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

M0.7R400 | 0.7 | 7 500 | 437.4 | 396.4 | 0.011 | 1.082 | 416 | 176 | 208 | 10.515 | 0.855 | 0.0180 | 6.309 |

M0.7R600 | 0.7 | 11 750 | 652.1 | 591.1 | 0.010 | 1.082 | 608 | 256 | 320 | 10.725 | 0.875 | 0.0124 | 6.113 |

M1.7R200 | 1.7 | 4 500 | 321.6 | 196.6 | 0.057 | 1.483 | 304 | 128 | 160 | 10.579 | 0.867 | 0.0247 | 6.030 |

M1.7R400 | 1.7 | 10 000 | 663.1 | 406.3 | 0.053 | 1.481 | 800 | 246 | 400 | 8.288 | 0.926 | 0.0129 | 4.973 |

M1.7R600 | 1.7 | 15 500 | 971.7 | 595.8 | 0.050 | 1.480 | 896 | 384 | 480 | 10.845 | 0.868 | 0.0082 | 6.073 |

M3.0R200 | 3.0 | 7 500 | 649.9 | 208.3 | 0.131 | 2.487 | 608 | 256 | 320 | 10.689 | 0.872 | 0.0124 | 6.093 |

M3.0R400 | 3.0 | 15 000 | 1232.5 | 395.5 | 0.123 | 2.486 | 1152 | 480 | 576 | 10.699 | 0.880 | 0.0066 | 6.419 |

M3.0R600 | 3.0 | 24 000 | 1876.1 | 600.7 | 0.116 | 2.491 | 1728 | 416 | 896 | 10.857 | 0.849 | 0.0093 | 6.282 |

M4.0R200 | 4.0 | 10 000 | 1017.5 | 202.8 | 0.189 | 3.637 | 1260 | 384 | 644 | 8.075 | 0.909 | 0.0082 | 4.740 |

Case name . | Ma_{b}
. | Re_{h}
. | Re_{τ}
. | $Re\tau \u2217$ . | −B
. _{q} | T/_{c}T
. _{w} | n
. _{x} | n
. _{y} | n
. _{z} | Δx^{+}
. | $\Delta ymin+$ . | Δy_{max}/h
. | Δz^{+}
. |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

M0.7R400 | 0.7 | 7 500 | 437.4 | 396.4 | 0.011 | 1.082 | 416 | 176 | 208 | 10.515 | 0.855 | 0.0180 | 6.309 |

M0.7R600 | 0.7 | 11 750 | 652.1 | 591.1 | 0.010 | 1.082 | 608 | 256 | 320 | 10.725 | 0.875 | 0.0124 | 6.113 |

M1.7R200 | 1.7 | 4 500 | 321.6 | 196.6 | 0.057 | 1.483 | 304 | 128 | 160 | 10.579 | 0.867 | 0.0247 | 6.030 |

M1.7R400 | 1.7 | 10 000 | 663.1 | 406.3 | 0.053 | 1.481 | 800 | 246 | 400 | 8.288 | 0.926 | 0.0129 | 4.973 |

M1.7R600 | 1.7 | 15 500 | 971.7 | 595.8 | 0.050 | 1.480 | 896 | 384 | 480 | 10.845 | 0.868 | 0.0082 | 6.073 |

M3.0R200 | 3.0 | 7 500 | 649.9 | 208.3 | 0.131 | 2.487 | 608 | 256 | 320 | 10.689 | 0.872 | 0.0124 | 6.093 |

M3.0R400 | 3.0 | 15 000 | 1232.5 | 395.5 | 0.123 | 2.486 | 1152 | 480 | 576 | 10.699 | 0.880 | 0.0066 | 6.419 |

M3.0R600 | 3.0 | 24 000 | 1876.1 | 600.7 | 0.116 | 2.491 | 1728 | 416 | 896 | 10.857 | 0.849 | 0.0093 | 6.282 |

M4.0R200 | 4.0 | 10 000 | 1017.5 | 202.8 | 0.189 | 3.637 | 1260 | 384 | 644 | 8.075 | 0.909 | 0.0082 | 4.740 |

The resulting grid sizes and grid-spacings are listed in Table IV. In viscous wall-units, the grid-spacings in the wall-parallel directions are commensurate with typical DNS grids while the wall-normal grid-spacing in the first grid-point is larger than usual; this choice is made to minimize the acoustic time step restriction. The grid-convergence was studied thoroughly. The most challenging case is M4.0R200, since the local viscous length scale increases with temperature; therefore, it is the smallest near the wall in the most strongly cooled flow. The grid-convergence for this case is illustrated in Fig. 9. All other cases converge at least equally well. As a side point, it is interesting to note the drastically higher computational cost for the strongly cooled cases, due to the spatially varying viscous length scale.

Label . | n
. _{x} | n
. _{y} | n
. _{z} | Δx^{+}
. | $\Delta ymin+$ . | Δy_{max}/h
. | Δz^{+}
. |
---|---|---|---|---|---|---|---|

A | 620 | 272 | 272 | 16.467 | 1.289 | 0.0116 | 11.261 |

B | 940 | 272 | 480 | 10.799 | 1.281 | 0.0116 | 6.344 |

C | 760 | 384 | 380 | 13.571 | 0.921 | 0.0082 | 8.143 |

D | 940 | 384 | 480 | 10.789 | 0.906 | 0.0082 | 6.339 |

E | 1260 | 384 | 644 | 8.075 | 0.909 | 0.0082 | 4.740 |

Label . | n
. _{x} | n
. _{y} | n
. _{z} | Δx^{+}
. | $\Delta ymin+$ . | Δy_{max}/h
. | Δz^{+}
. |
---|---|---|---|---|---|---|---|

A | 620 | 272 | 272 | 16.467 | 1.289 | 0.0116 | 11.261 |

B | 940 | 272 | 480 | 10.799 | 1.281 | 0.0116 | 6.344 |

C | 760 | 384 | 380 | 13.571 | 0.921 | 0.0082 | 8.143 |

D | 940 | 384 | 480 | 10.789 | 0.906 | 0.0082 | 6.339 |

E | 1260 | 384 | 644 | 8.075 | 0.909 | 0.0082 | 4.740 |

## REFERENCES

_{τ}= 590

^{+}≈ 2000

_{τ}= 5200