# Improved Wall-Normal Derivative Formulae for Anisotropic Adaptive Simplex-
Element Grids

Hiroaki Nishikawa
National Institute of Aerospace, Hampton, VA 23666, USA Research Fellow
(hiro@nianet.org), 100 Exploration Way, Hampton, VA 23666 USA, Associate
Fellow AIAA

###### Abstract

In this paper, we explore methods for computing wall-normal derivatives used
for calculating wall skin friction and heat transfer over a solid wall in
unstructured simplex-element (triangular/tetrahedral) grids generated by
anisotropic grid adaptation. Simplex-element grids are considered as efficient
and suitable for automatic grid generation and adaptation, but present a
challenge to accurately predict wall-normal derivatives. For example, wall-
normal derivatives computed by a simple finite-difference approximation, as
typically done in practical fluid-dynamics simulation codes, are often
contaminated with numerical noise. To address this issue, we propose an
improved method based on a common step-length for the finite-difference
approximation, which is otherwise random due to grid irregularity and thus
expected to smooth the wall-normal derivative distribution over a boundary.
Also, we consider using least-squares gradients to compute the wall-normal
derivatives and discuss their possible improvements. Numerical results show
that the improved methods greatly reduce the noise in the wall-normal
derivatives for irregular simplex-element grids.

##  1 Introduction

Towards fully automated computational fluid dynamics (CFD) simulations, there
is an increasing interest in developing robust and accurate discretizations
and solvers for unstructured simplex-element grids with anisotropic grid
adaptation
[WhiteNishikawaBaurle_scitech2020,Kleb_etal_aiaa2019-2948,Nishikawa_scitech2020,nishikawa_centroid:JCP2020,uns_grid_adaptation_aiaa2018-1103,alauzet-
loseille-decade-aniso-adapt-cfd]. While fully-automated CFD simulations have
recently been successfully demonstrated for practical three-dimensional
problems [Kleb_etal_aiaa2019-2948], improvements are still desired in the
discretization method especially in terms of accuracy. For example,
computations of derivatives such as the viscous/heat fluxes and the vorticity
are known to suffer from numerical noise on highly irregular and skewed
simplex-element grids [liu_nishikawa_aiaa2016-3969]. Impact of numerical noise
may be reduced in integrated quantities (e.g., drag coefficient), but when
accuracy is required for a pointwise quantity, e.g., heating rate at a
stagnation point, numerical noise can be a serious problem, causing a large
error and a great degree of uncertainty in the prediction. In order to develop
a reliable automated CFD solver, it is necessary to be able to accurately
predict pointwise derivatives, especially at wall boundaries, on irregular
anisotropic simplex-element grids.

There have been efforts to improve derivative accuracy on unstructured grids:
derivative variables introduced in target equations
[nishikawa_hyperbolic_poisson:jcp2020,liu_nishikawa_aiaa2016-3969], improved
least-squares methods
[NovelGradStencil:CF2018,SozerChristophCetin:AIAA2014,ShimaKitamuraHaga_AIAAJ2013],
and implicit gradient methods [WangRenPanLi:JCP2017,nishikawa_igg:JCP2018].
All these methods are promising for improving derivative accuracy, but they
are not specifically designed to accurately estimate derivatives at a boundary
and thus further improvements may be achieved by developing a more specific
technique. In this work, we focus on wall-normal derivatives and explore
various methods for accurately estimating the wall-normal derivatives relevant
to the viscous stresses and heat fluxes at a wall, as a post-processing step
for a given numerical solution on adaptive simplex-element grids. A popular
technique used in practical CFD codes for computing the wall-normal
derivatives is a one-sided finite-difference formula applied in the wall-
normal direction with the numerical solution in a nearby node/cell and a
boundary condition value. For example, it provides an approximation to
$\partial u/\partial n$ that dominates the viscous stress at a wall, where $u$
is a wall-parallel velocity components and $n$ is the coordinate normal to the
wall. It can provide an accurate approximation on structured grids, but will
introduce numerical noise on irregular simplex-element grids
[WhiteNishikawaBaurle_scitech2020]. We conjecture that the numerical noise in
the derivative is due to the irregularity of the spacing used in the finite-
difference formula. That is, the distance between the center of a cell
adjacent to a boundary and a wall-boundary face centroid varies along a
boundary in adaptive simplex-element grids. Then, we propose to keep the same
distance for the finite-difference formula applied at all wall-boundary faces.
Numerical experiments show that the modified finite-difference method greatly
improves the skin friction distribution over a flat plate in a laminar flow,
where the distance is determined as $\eta=0.5$ at all boundary faces, where
$\eta$ is a normalized coordinate in the Blasius boundary-layer solution. In
order to extend this technique to more general flows over complex geometries,
it is necessary to develop a method for estimating a universal distance for
the finite-difference formula.

Another post-processing technique can be devised based on gradients computed
by a least-squares method at nodes or cells. Following a recent study
[WhiteNishikawaBaurle_scitech2020], where heating rate prediction over a flat
plate in a hypersonic turbulent flow has been shown to be greatly improved by
the face-averaged nodal-gradient (F-ANG) approach applied to the finite-volume
discretization [WhiteNishikawaBaurle_scitech2020,NishikawaWhite_FANG:jcp2020],
we employ the F-ANG approach in this study, but only consider the post-
processing technique for estimating the wall-normal derivatives for a given
numerical solution. Discussions on the impact of the gradient approach (used
in the discretization) on wall heating prediciton are given in a companion
paper [NishikawaWhite_scitech2021-xxxx]. In the F-ANG method, gradients are
computed and stored at nodes while numerical solutions are stored at cells.
Therefore, the wall-normal derivatives are already available at wall nodes.
However, these derivatives can be noisy for irregular anisotropic simplex-
element grids. A simple method for reducing the noise would be to average the
nodal gradients, and there are two possible techniques: face and cell
averages. We compare these averaged gradients for estimating the wall-normal
derivatives. Numerical results indicate that the cell-averaged nodal-gradients
can greatly reduce the noise in the wall-normal derivatives, but not as much
as the improved finite-difference formula.

##  2 Finite-Volume Discretization

We consider a cell-centered finite-volume discretization for the Navier-Stokes
equations integrated over a computational cell $j$ (see Figure 1) with the
midpoint rule:

$\displaystyle{\bf Res}_{j}=\sum_{k\in\\{k_{j}\\}}{\bf\Phi}_{jk}A_{jk},$ (2.1)

where $\\{k_{j}\\}$ is a set of neighbors of the cell $j$, $A_{jk}$ is the
length of the face across $j$ and $k$, and ${\bf\Phi}_{jk}$ is a numerical
flux. In this work, the Roe [Roe_JCP_1981] flux is used for the inviscid
terms, and the alpha-damping flux [nishikawa:AIAA2010] for the viscous terms.
The solution values are stored in the primitive variables ${\bf w}=(\rho,{\bf
v},T)$, at cells as point values at the centroid [nishikawa_centroid:JCP2020],
where $\rho$ is the density, ${\bf v}$ is the velocity vector, and $T$ is the
temperature. Second-order accuracy is achieved by the linearly-exact flux
quadrature (i.e., the midpoint rule) and the numerical flux evaluated with
linearly reconstructed solutions. In this work, we focus on the F-ANG approach
[WhiteNishikawaBaurle_scitech2020,NishikawaWhite_FANG:jcp2020], which has been
shown to yield significantly more accurate wall-derivatives than a
conventional finite-volume scheme, but still involves mild oscillations in the
derivatives on irregular simplex-element grids. In the F-ANG approach, we
compute the solution gradients at nodes using solution values stored at cells
by a linear least-squares method. Then, the nodal gradients are used in the
linear reconstruction, averaged over the nodes of a face:

$\displaystyle{\bf w}_{L}={\bf w}_{j}+\nabla{\bf w}_{f}\cdot\Delta{\bf
x}_{jm},\quad{\bf w}_{R}={\bf w}_{k}+\nabla{\bf w}_{f}\cdot\Delta{\bf
x}_{km},$ (2.2)

where $\nabla{\bf w}_{f}$ denotes the face-averaged gradient:

$\displaystyle\nabla{\bf w}_{f}=\frac{1}{2}\left(\nabla{\bf
w}_{jk}^{\ell}+\nabla{\bf w}_{jk}^{r}\right),$ (2.3)

where the superscripts $\ell$ and $r$ denote the left and right nodes of the
face $[j,k]$ (see Figure 1). This approach has various advantages especially
for simplex-element grids: e.g., no interpartition is required for gradients,
the residual stencil is greatly reduced. See
[WhiteNishikawaBaurle_scitech2020,NishikawaWhite_FANG:jcp2020] for further
details.

Figure 1: Stencil for cell-centered finite-volume discretization.

##  3 Methods for Computing Wall-Normal Derivatives

In this section, we describe methods for computing a wall-normal derivative,
taking the velocity derivative $\partial u/\partial n$ as an example, where
$u$ is the wall-parallel component of the velocity and $n$ denotes the
coordinate normal to the wall. We assume that the value at a wall $u_{b}$ is
known, e.g., $u_{b}=0$ for a no-slip condition. The wall-normal derivative
will be computed at a boundary node or at a boundary face, depending on the
method.

Figure 2: Geometry of an irregular triangular grid over a boundary. Black
filled circles are boundary nodes; smaller circles are the centroids of
triangles adjacent to the boundary.

###  3.1 Nodal gradients

The wall-normal derivative is computed at a boundary node $i$ by projecting
the nodal gradient $\nabla u_{i}$ in the direction of wall normal (see Figure
2):

$\displaystyle\left.\frac{\partial u}{\partial n}\right|_{i}=\nabla
u_{i}\cdot\hat{\bf n}_{i},$ (3.1)

where $\hat{\bf n}_{i}$ denotes a unit vector at the node $i$ normal to the
wall pointing towards the interior domain. This is perhaps the simplest method
since the nodal gradients are already available in the F-ANG method.

###  3.2 Face-averaged nodal gradients

The wall-normal derivative is computed at a face $j$ with a face-averaged
gradient:

$\displaystyle\left.\frac{\partial u}{\partial
n}\right|_{j}=\overline{\nabla}u_{j}^{face}\cdot\hat{\bf n}_{j},$ (3.2)

where $\overline{\nabla}u_{j}^{face}$ is the average of the nodal gradients
over the boundary face $j$. In Figure 2, $\overline{\nabla}u_{j}^{face}$ is
obtained by averaging the nodal gradients at the left and right nodes of the
boundary face $j$.

###  3.3 Cell-averaged nodal gradients

The wall-normal derivative is computed at a face $j$ with a cell-averaged
gradient:

$\displaystyle\left.\frac{\partial u}{\partial
n}\right|_{j}=\overline{\nabla}u_{j}^{cell}\cdot\hat{\bf n}_{j},$ (3.3)

where $\overline{\nabla}u_{j}^{cell}$ is the average of the nodal gradients
over a cell adjacent to the face $j$. In Figure 2,
$\overline{\nabla}u_{j}^{cell}$ is obtained by averaging the nodal gradients
at the three nodes of the triangular cell $c$.

(a) Finite-difference I.

(b) Finite-difference II.

(c) Finite-difference III.

Figure 3: Geometries for the finite-difference wall-normal derivative
formulae.

###  3.4 Finite-difference I

The wall-normal derivative can be computed at a face $j$ by the following
finite-difference formula:

$\displaystyle\left.\frac{\partial u}{\partial
n}\right|_{j}=\frac{u_{c}-u_{b}}{({\bf x}_{c}-{\bf x}_{j})\cdot\hat{\bf
n}_{j}},$ (3.4)

where $u_{c}$ is a solution value stored at the cell $c$ adjacent to the
boundary face $j$, $u_{b}$ is a known boundary value (e.g., from a boundary
condition), ${\bf x}_{c}$ is the centroid position of the cell $c$, ${\bf
x}_{j}$ is the centroid of the boundary face $j$. See Figure 3(a). This
formula estimates the derivative at a point where the cell-center ${\bf
x}_{c}$ is projected on the boundary, not at ${\bf x}_{j}$.

###  3.5 Finite-difference II

The wall-normal derivative can be estimated at the face centroid by

$\displaystyle\left.\frac{\partial u}{\partial
n}\right|_{j}=\frac{u_{c^{\prime}}-u_{b}}{({\bf x}_{c}-{\bf
x}_{j})\cdot\hat{\bf n}_{j}},$ (3.5)

where $u_{c^{\prime}}$ is a solution extrapolated from the cell center ${\bf
x}_{c}$ to the point right above the face centroid (see Figure 3(b)):

$\displaystyle u_{c^{\prime}}=u_{c}+\nabla u_{c}\cdot({\bf
x}_{c^{\prime}}-{\bf x}_{c}),\quad{\bf x}_{c^{\prime}}={\bf x}_{j}+[({\bf
x}_{c}-{\bf x}_{j})\cdot\hat{\bf n}_{j}]\hat{\bf n}_{j}.$ (3.6)

This approximation is exact for linear functions at the face centroid.

###  3.6 Finite-difference III

In the previous methods, the finite-difference formula is applied with varying
step-length as the distance between the cell center and the face center in the
normal direction changes randomly on irregular simplex-element grids. Hence,
the methods are strongly dependent on the grid geometry, which is not suitable
for adaptive simplex-element grids. To minimize the dependence on the grid
geometry, we generalize the previous formula as

$\displaystyle\left.\frac{\partial u}{\partial
n}\right|_{j}=\frac{u_{p}-u_{b}}{\overline{h}},$ (3.7)

where $u_{p}$ is a solution extrapolated from the cell center ${\bf x}_{c}$ to
the point at a certain distance $\overline{h}$ from the face center in the
wall normal direction (see Figure 3(c)):

$\displaystyle u_{p}=u_{c}+\nabla u_{c}\cdot({\bf x}_{p}-{\bf
x}_{c}),\quad{\bf x}_{p}={\bf x}_{j}+\overline{h}\,\hat{\bf n}_{j},$ (3.8)

where $\overline{h}$ is a global constant or varies along a wall but constant
in a transformed normal coordinate. This approximation is also exact for
linear functions at the face centroid.

This formula does not depend strongly on the cell centroids of the adjacent
cells. Whether in the physical or transformed coordinate, the above formula
keeps the same step-length of the finite-difference formula over a wall and
thus is expected to produce a smoother derivative (and its error)
distribution.

In this work, we investigate the effects of the use of a common step-length
$\overline{h}$ and explore estimation techniques of $\overline{h}$ for various
viscous-flow problems on adaptive grids. Here, we provide preliminary results
for a laminar flat-plate computation with the height $\overline{h}$ determined
as described below.

###  3.7 Finite-difference with face-area-weighted centroids

Finite-difference III requires a common length to be defined. For a laminar
flow over a flat plate, a normalized wall-normal distance, as we will describe
later, has been found to serve well for subsonic flows. However, such an
estimate is not applicable to high-speed flows and practical turbulent flows
over complex geometries. As a practical technique, we consider the face-area-
weighted centroid [nishikawa_centroid:JCP2020], which could reduce the
centroid-height variation along a boundary. The resulting method is similar to
the finite-difference II but ${\bf x}_{c^{\prime}}$ is replaced by the face-
area-weighted centroid:

$\displaystyle{\bf
x}_{c^{\prime}}=\frac{\displaystyle\sum_{k\in\\{k_{j}\\}}\hat{A}_{jk}^{p}{\bf
x}_{jk}}{\displaystyle\sum_{k\in\\{k_{j}\\}}\hat{A}_{jk}^{p}},\quad\hat{A}_{jk}=\frac{{A}_{jk}}{\displaystyle\max_{k\in\\{k_{j}\\}}{A}_{jk}},$
(3.9)

where${A}_{jk}$ is the area (length in two dimensions) of the face across $j$
and $k$, ${\bf x}_{jk}$ is the face centroid, and $p=2$.

##  4 Numerical Results

###  4.1 Two-dimensional laminar flat plate at $M_{\infty}=0.15$,
$Re_{\infty}=10^{6}$

For a laminar flat plate, we consider the normalized wall-normal coordinate
$\eta$:

$\displaystyle\eta=\frac{y}{x}\sqrt{Re_{x}},\quad
Re_{x}=\frac{\rho_{\infty}U_{\infty}x}{\mu_{\infty}},$ (4.1)

where $x$ is the distance from the leading edge, $\rho_{\infty}$,
$U_{\infty}$, $\mu_{\infty}$ are free stream density, flow speed, and
viscosity. For the finite-difference III, we fix the height to be $\eta=0.5$,
which is translated into the $y$-coordinate as

$\displaystyle\overline{h}=\eta\sqrt{\frac{x_{j}}{Re_{\infty}}},$ (4.2)

where $\eta=0.5$, $x_{j}$ is the $x$-coordinate of the face centroid, and
$Re_{\infty}$ is the free stream Reynolds number per unit length. Note that we
compute the wall-normal velocity derivative at boundary face centroids and
thus $x_{j}$ will not be zero. As can be seen from the formula, the height
$\overline{h}$ changes in the physical coordinate parabolically but remains
constant in the normalized wall-normal coordinate $\eta$. In the results, this
method is denoted by FD-III($\eta=0.5$). For a more general flow problem, this
approach may not be suitable. Therefore, we also consider using a fixed length
over the wall in the physical space:

$\displaystyle\overline{h}=\eta\sqrt{{Re_{\infty}}},$ (4.3)

which corresponds to using the estimated distance with $\eta=0.5$ at
$x_{j}=1.0$ everywhere. In this case, $\overline{h}$ is a global constant.

An adaptive grid was generated by the open-source tool refine
[park_darmofal:AIAA2008-917] developed by Mike Park at NASA Langley Research
Center and available at https://github.com/nasa/refine. Adaptation was
performed with the multiscale metric technique to control interpolation error
in L2-norm [ugawg-aiaa-verification] with Mach Hessian computed at nodes (by a
least-squares method) from the numerical solution obtained at cells with the
F-ANG finite-volume solver for the Navier-Stokes equations. The resulting
adaptive grid is shown in Figures 4(a) and 4(b). As can be seen, it is a fully
irregular grid towards the flat plate located at the bottom in $x\in[0,2]$.
Note that we already have a numerical solution on this grid as shown by Mach
contours in Figure 4(b). Then, the skin friction is computed at the flat plate
as

$\displaystyle C_{fx}=(\tau_{xx},\tau_{xy})\cdot\hat{\bf
n}\frac{2}{M_{\infty}^{2}}=\mu\left(\frac{2}{3}\frac{\partial u}{\partial
x}-\frac{1}{3}\frac{\partial v}{\partial y},\frac{\partial u}{\partial
y}+\frac{\partial v}{\partial x}\right)\cdot\hat{\bf
n}\frac{2}{M_{\infty}^{2}}=\mu\left(\frac{\partial u}{\partial
y}+\frac{\partial v}{\partial x}\right)\frac{2}{M_{\infty}^{2}},$ (4.4)

where $u$ and $v$ are the $x$\- and $y$-components of the velocity, the factor
$1/M_{\infty}^{2}$ comes from the nondimensionalization (i.e., the velocity is
nondimensionaliezd by the free stream speed of sound), and $\hat{\bf n}$ is
either $\hat{\bf n}_{i}$ or $\hat{\bf n}_{j}$, depending on where the
gradients are computed. The term $\partial v/\partial x$ is ignored in the
finite-difference methods, but retained in the LSQ gradient methods.

Results are shown in Figures 4(c)-4(i), where numerical skin-friction
distributions are compared with that of the Blasius solution. All numerical
skin-friction distributions are close to the theoretical distribution on
average, but significant differences are observed in the noise level. First,
the FD-I and FD-II methods produce noisy skin-friction distributions as shown
in Figures 4(c) and 4(d). On the other hand, as expected, the proposed methods
produce much smoother distributions as shown in Figures 4(e) and 4(f). Note
that the choice of $\overline{h}$ has no major impact on the skin-friction
distribution: Figure 4(e) for $\eta=0.5$, and Figure 4(f) for a global
constant corresponding to $\eta=0.5$ and $x=1$. These results indicate that it
is the uniformness of the step-length, whether in $\eta$ or $y$, that affects
the smoothness of the skin friction distribution. This is encouraging since
the method can be extended to more general problems by developing a method for
estimating a single value of the step-length $\overline{h}$ in a given
physical problem.

For the LSQ gradient methods, the direct use of the nodal gradients leads to a
noisy skin-friction distribution as shown in Figure 4(g). A slightly smoother
distribution is obtained with the face-averaged nodal gradients as shown in
Figure 4(h). The smoothest distribution is obtained with the cell-averaged
nodal gradients as shown in Figure 4(i). The results indicate that averaging
with more nodal gradients leads to a smoother distribution, but not as smooth
as those obtained with the finite-difference III. Note that the gradients are
linearly exact no mater how they are averaged, and the averaging operator acts
as a kind of filtering to remove high-frequency contents from the underlying
gradient approximation.

(a) Adaptive grid (3457 nodes).

(b) Zoomed-in view of the grid (with a scaled $y$-coordinate) and Mach
contours.

(c) FD-I.

(d) FD-II.

(e) FD-III ($\eta=0.5$).

(f) FD-III ($\eta=0.5$, $x=1$).

(g) Nodal gradients.

(h) Face-averaged gradients.

(i) Cell-averaged gradients.

Figure 4: Results for a laminar flow over a flat plate at $M_{\infty}=0.15$
and $Re_{\infty}=10^{6}$ on an adaptive triangular grid. The flat plate is
located at $y=0$ in $x\in[0,2]$. The skin friction coefficient $C_{fx}$ is
plotted at wall face centers, except for the figure (g).

##  5 Concluding Remarks

We have proposed improved wall-normal derivative formulae for adaptive
anisotropic simplex-element grids, based on the simple idea of unifying the
finite-difference distance over a wall. The improved formulae have been
demonstrated for a laminar flow over a flat plate to reduce the amount of
numerical noise in the skin friction distribution. In the future work, these
formulae will have to be extended to three dimensions and to turbulent flows.

## Acknowledgments

This work was supported by the Hypersonic Technology Project, through the
Hypersonic Airbreathing Propulsion Branch of the NASA Langley Research Center,
under Contract No. 80LARC17C0004. The author would like to thank Mike Park
(NASA Langley Geometry Laboratory) for his assistance with the use of refine.

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$\|\vec{b}-\vec{g}\|_{\infty}<KL^{1/4}$, we have

$\mathbb{P}(\mathcal{E}_{\vec{b}})=(1+O(K^{2}L^{-1/4}\log
L))\mathbb{P}(\mathcal{E}_{\vec{g}})+O(\exp(-cK^{2}M^{-1}L^{1/2}\log L)).$

Now for each $i=1,\ldots,\lfloor L^{1/4}\rfloor$, and any
$\vec{b}\in\Theta_{iDL^{1/4}}$, $\vec{g}\in\Theta_{DL^{1/4}}$, we can find a
sequence of vectors in $\Theta$,
$\vec{b}=\vec{b}^{(0)},\vec{b}^{(1)},\ldots,\vec{b}^{(i)}=\vec{g}$, with
$\|\vec{b}^{(j-1)}-\vec{b}^{(j)}\|_{\infty}\leq DL^{1/4}$ for each
$j=1,\ldots,i$. Then (12.9) implies that for each $j$

$\mathbb{P}(\mathcal{E}_{\vec{b}^{(j-1)}})<(1+CD^{2}L^{-1/4}\log
L)\mathbb{P}(\mathcal{E}_{\vec{b}^{(j)}})+C\exp(-cD^{2}M^{-1}L^{1/2}\log L),$

and, since $1+x\leq\exp(x)$,

$\mathbb{P}(\mathcal{E}_{\vec{b}})<\exp(CiD^{2}L^{-1/4}\log
L)\left(\mathbb{P}(\mathcal{E}_{\vec{g}})+C\exp(-cD^{2}M^{-1}L^{1/2}\log
L)\right).$

In other words, we have (for each $i=1,\ldots,\lfloor L^{1/4}\rfloor$)

$\max_{\vec{b}\in\Theta_{iDL^{1/4}}}\mathbb{P}(\mathcal{E}_{\vec{b}})<\exp(CiD^{2}L^{-1/4}\log
L)\left(\min_{\vec{b}\in\Theta_{DL^{1/4}}}\mathbb{P}(\mathcal{E}_{\vec{b}})+C\exp(-cD^{2}M^{-1}L^{1/2}\log
L)\right).$

Therefore, by plugging this into (12.16) (with each
$\mathbb{P}(\mathcal{E}_{\vec{b},+})$ replaced by
$\mathbb{P}(\mathcal{E}_{\vec{b}})$ using (12.14)) we have

$\mathbb{P}(\mathcal{E}_{+}\mid\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h})<(1+C\exp(-cD^{2}))\max_{\vec{b}\in\Theta_{DL^{1/4}}}\mathbb{P}(\mathcal{E}_{\vec{b}})+C\exp(-cD^{2}M^{-1}L^{1/2}\log
L).$

Also, (12.15) (with each $\mathbb{P}(\mathcal{E}_{\vec{b},-})$ replaced by
$\mathbb{P}(\mathcal{E}_{\vec{b}})$ using (12.14)) and Lemma 12.9 imply that

$\mathbb{P}(\mathcal{E}_{-}\mid\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h})>(1-C\exp(-cD^{2}))\max_{\vec{b}\in\Theta_{DL^{1/4}}}\mathbb{P}(\mathcal{E}_{\vec{b}})-C\exp(-cD^{2}M^{-1}L^{1/2}\log
L).$

Since that (as we have seen above) (12.12) is lower bounded by
$c\exp(-CL^{1/2})$ ($\beta=1$) or $c$ ($\beta=\infty$) by Proposition 9.1, and
the RHS of (12.12) is in the interval (12.13), we conclude that (12.12) equals
$(1+o(1))\max_{\vec{b}\in\Theta_{DL^{1/4}}}\mathbb{P}(\mathcal{E}_{\vec{b}})$,
which is independent of $\vec{x}$. ∎

It remains to prove the comparison lemma on Brownian bridges.

###### Proof of Lemma 12.9.

We note that the events $\mathcal{E}_{\vec{b}}$ and $\mathcal{E}_{\vec{g}}$
are measurable with respect to the processes $B^{\dagger,i,\vec{b}}$ in the
interval $I_{M,L}$. Therefore, conditional on that each
$B^{\dagger,i,\vec{b}}$ and $B^{\dagger,i,\vec{g}}$ are the same at $\pm
ML^{-1/2}\log L$, the conditional probabilities of $\mathcal{E}_{\vec{b}}$ and
$\mathcal{E}_{\vec{g}}$ would be the same.

We let $\mathrm{BdyCtrl}$ be the event where for each $\dagger=\mathrm{top}$,
$i=1,\ldots,k-1$, and $\dagger=\mathrm{bot}$, $i=2,\ldots,k$, we have

$|B^{\dagger,i,\vec{b}}(ML^{-1/2}\log
L)-b^{\dagger,\mathrm{R}}_{i}\theta^{-1}ML^{-1/2}\log L+2M\log L|<K\log L,$
$|B^{\dagger,i,\vec{b}}(-ML^{-1/2}\log
L)-b^{\dagger,\mathrm{L}}_{i}\theta^{-1}ML^{-1/2}\log L+2M\log L|<K\log L,$
$|B^{\dagger,i,\vec{g}}(ML^{-1/2}\log
L)-b^{\dagger,\mathrm{R}}_{i}\theta^{-1}ML^{-1/2}\log L+2M\log L|<K\log L,$
$|B^{\dagger,i,\vec{g}}(-ML^{-1/2}\log
L)-b^{\dagger,\mathrm{L}}_{i}\theta^{-1}ML^{-1/2}\log L+2M\log L|<K\log L.$

It follows from Gaussian tail bounds (Lemma 2.22) (and noting that
$\|\vec{b}-\vec{g}\|\leq KL^{1/4}$) that

$\displaystyle\mathbb{P}\left(\mathrm{BdyCtrl}^{c}\right)\leq\exp\left(-cK^{2}M^{-1}L^{1/2}\log
L\right).$

It then suffices to control the ratio of the probability densities of
$B^{\dagger,i,\vec{b}}(\pm ML^{-1/2}\log L)$ and $B^{\dagger,i,\vec{g}}(\pm
ML^{-1/2}\log L)$, in the interval $-ML^{1/2}\log L+[-K\log L,K\log L]$. For
this, note that $B^{\dagger,i,\vec{b}}(\pm ML^{-1/2}\log L)$ and
$B^{\dagger,i,\vec{g}}(\pm ML^{-1/2}\log L)$ are Gaussian random variables
with the same variance of order $ML^{-1/2}\log L$, and mean differ by $\leq
K\theta^{-1}ML^{-1/4}\log L$. Therefore the density ratio is

$\exp(O((K\log L)(K\theta^{-1}ML^{-1/4}\log L)(ML^{-1/2}\log
L)^{-1}))=1+O(K^{2}L^{-1/4}\log L).$

Therefore the conclusion follows. ∎

##  Appendix A Weak convergence lemma

###### Proof of Lemma 11.2.

Fix $M>0$. We then have that

$\max_{\|\vec{x}\|_{\infty},\|\vec{y}\|_{\infty}\leq
M,\|\vec{x}-\vec{y}\|_{\infty}\leq\epsilon}\left|\frac{f(\vec{x})}{f(\vec{y})}-1\right|\to
1.$ (A.1)

This follows from the continuity and strict positivity of $f$ combined with
the compactness of $[-M,M]^{d}$. Now we take any $\delta>0$, and
$\vec{x},\vec{y}\in[-M+\delta,M-\delta]^{d}$. For any $\varepsilon$ such that
$\delta\varepsilon^{-1}\in\mathbb{N}$, by splitting
$\vec{x}+[-\delta,\delta)^{d}$ into $(\delta\varepsilon^{-1})^{d}$ many
translations of $[-\varepsilon,\varepsilon)^{d}$, applying (11.1) across them,
and using (A.1), we have

$\frac{\mathbb{P}\left(\vec{X}_{\varepsilon}\in\vec{x}+[-\delta,\delta)^{d}\right)}{\mathbb{P}\left(\vec{X}_{\varepsilon}\in\vec{y}+[-\delta,\delta)^{d}\right)}\to\frac{\int_{\vec{x}+[-\delta,\delta)^{d}}f(\vec{z})\,\mathrm{d}\vec{z}}{\int_{\vec{y}+[-\delta,\delta)^{d}}f(\vec{z})\,\mathrm{d}\vec{z}},$

as $\epsilon\to 0$. Then by the continuity of $f$, and that
$\vec{X}_{\epsilon}\to\vec{X}$ in distribution, we conclude that

$\frac{\mathbb{P}\left(\vec{X}\in\vec{x}+[-\delta,\delta)^{d}\right)}{\mathbb{P}\left(\vec{X}\in\vec{y}+[-\delta,\delta)^{d}\right)}=\frac{\int_{\vec{x}+[-\delta,\delta)^{d}}f(\vec{z})\,\mathrm{d}\vec{z}}{\int_{\vec{y}+[-\delta,\delta)^{d}}f(\vec{z})\,\mathrm{d}\vec{z}}.$

We note that by sending $M\to\infty$, this holds for arbitrary
$\vec{x},\vec{y}\in\mathbb{R}^{d}$. Then by the integrability of $f$, we get

$\mathbb{P}\left(\vec{X}\in\vec{x}+[-\delta,\delta)^{d}\right)=\frac{\int_{\vec{x}+[-\delta,\delta)^{d}}f(\vec{z})\,\mathrm{d}\vec{z}}{\int_{\mathbb{R}^{d}}f(\vec{z})\,\mathrm{d}\vec{z}}.$

Then the conclusion follows by sending $\delta\to 0$. ∎

##  Appendix B Tent Brownian comparison and estimates

In this appendix, we provide the proofs of Lemma 2.18, Lemma 2.19, and Lemma
2.20.

###### Proof of Lemma 2.18.

We wish to prove Lemma 2.18 from Theorem 2.17, by applying the tail comparison
of Theorem 4.1. However, the proof of Theorem 4.1 we give uses Lemma 2.18, and
we next explain how circular arguments are avoided.

First, recall the notation $L_{M}^{1/2}=(L-M)^{1/2}$ from the proof of Theorem
4.1. That proof used Lemma 2.18 in the special case of

$\displaystyle\mathbb{P}\left(\max_{x\in\\{-L_{M}^{1/2},L_{M}^{1/2}\\}}\Bigl{|}\hat{\mathfrak{h}}^{\beta}_{t,1}(x)-L+2L^{1/2}|x|\Bigr{|}>ML^{1/4}\
\Big{|}\ \hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L\right)<\exp(-cM^{2}),$

for $M=\log L$, i.e., the process is considered only at the two points $\pm
L_{M}^{1/2}$ and not on the entire interval $[-L^{1/2},L^{1/2}]$; this was
done in the proof of Theorem 4.1 right before (4.8). This application of Lemma
2.18 with this value of $M$ is the only one in the proof of Theorem 4.1. We
will show how to prove the above display, and then we will be allowed to use
Theorem 4.1 to prove Lemma 2.18.

Call the event in the previous display $A_{M,L}$. Take a large $C_{1}>0$, then
the previous display is bounded by

$\displaystyle\mathbb{P}\left(A_{M,L},\hat{\mathfrak{h}}^{\beta}_{t,1}(0)<L+C_{1}L^{1/4}\mid\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L\right)+\mathbb{P}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L+C_{1}L^{1/4}\mid\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L\right).$
(B.1)

The second term is upper bounded, using a trivial upper bound and then Theorem
2.13, by

$\displaystyle\frac{\mathbb{P}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L+C_{1}L^{1/4}\right)}{\mathbb{P}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L\right)}\leq\frac{\exp\left(-\frac{4}{3}(L+C_{1}L^{1/4})^{3/2}+CL^{3/4}\right)}{\exp\left(-\frac{4}{3}L^{3/2}-CL^{3/4}\right)}\leq\exp\left(-cL^{3/4}\right),$
(B.2)

as $C_{1}$ is large. The first term of (B.1) is upper bounded by

$\displaystyle\mathbb{P}\left(A_{M,L}\ \Big{|}\
\hat{\mathfrak{h}}^{\beta}_{t,1}(0)\in(L,L+C_{1}L^{1/4})\right)\leq\sup_{L^{\prime}\in(L,L+C_{1}L^{1/4})}\mathbb{P}\left(A_{M,L}\
\Big{|}\
\hat{\mathfrak{h}}^{\beta}_{t,1}(0)\in(L^{\prime},L^{\prime}+\mathrm{d}L^{\prime})\right).$

Now applying the Theorem 2.17 where we condition on the value of
$\hat{\mathfrak{h}}^{\beta}_{t,1}(0)$ yields that the final term is upper
bounded by $\exp(-cM^{2})$ when $0<M<cL^{3/4}$. Thus the bound on (B.1) is
$\exp(-cM^{2})+\exp(-cL^{3/4})\leq\exp(-cM^{2})$ when $M<L^{3/8}$, as is
certainly the case for $M=\log L$.

This establishes the needed bound on (B.1) and so we may now make use of
Theorem 4.1, as noted above. We now perform the same analysis as above but
with

$\displaystyle\tilde{A}_{M,L}=\left\\{\sup_{|x|\leq
L^{1/2}}\Bigl{|}\hat{\mathfrak{h}}^{\beta}_{t,1}(x)-L+2L^{1/2}|x|\Bigr{|}>ML^{1/4}\right\\}$

replacing $A_{M,L}$ and with $M^{2}L^{-1/2}$ replacing $C_{1}L^{1/4}$; the
only difference is that the bound in (B.2) now follows from Theorem 4.1
instead of Theorem 2.13 (and is now $\exp(-cM^{2})$ instead of
$\exp(-cL^{3/4})$). The remaining analysis is the same and yields an overall
bound on the LHS of (B.1) of $\exp(-cM^{2})$ for $0<M<cL^{3/4}$. ∎

###### Proof of Lemma 2.19.

Let $\mathcal{E}_{0}$ denote the event where
$\hat{\mathfrak{h}}^{\beta}_{t,1}(0)\in(L,L+\,\mathrm{d}L)$. Denote by
$\mathcal{E}_{1}$ the event

$\hat{\mathfrak{h}}^{\beta}_{t,2}(x)<-x^{2}+0.1L,\quad\forall
x\in[-\tfrac{1}{2}L^{1/2},\tfrac{1}{2}L^{1/2}],$

and by $\mathcal{E}_{2}$ the event where

$\hat{\mathfrak{h}}^{\beta}_{t,1}(-L^{1/2}/2),\hat{\mathfrak{h}}^{\beta}_{t,1}(L^{1/2}/2)>-0.1L.$

By Lemma 2.12 and Lemma 3.2, we have

$\mathbb{P}(\mathcal{E}_{1}\mid\mathcal{E}_{0})>\mathbb{P}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(x)<-x^{2}+0.1L,\;\forall
x\in[-\tfrac{1}{2}L^{1/2},\tfrac{1}{2}L^{1/2}]\right)>1-C\exp(-cL^{3/2}).$

By Theorem 2.17 we have
$\mathbb{P}(\mathcal{E}_{2}\mid\mathcal{E}_{0})>1-C\exp(-cL^{3/2})$.

We now assume the event
$\mathcal{E}_{0}\cap\mathcal{E}_{1}\cap\mathcal{E}_{2}$, and consider the
Radon-Nikodym derivative between the two sets of processes. Let
$\mathcal{F}=\mathcal{F}_{\mathrm{ext}}([-t^{2/3}L^{1/2}/2,t^{2/3}L^{1/2}/2])$
be the sigma-algebra generated by $\hat{\mathfrak{h}}^{\beta}_{t,1}$ on
$(-\infty,-L^{1/2}/2]\cup\\{0\\}\cup[L^{1/2}/2,\infty)$ and
$\hat{\mathfrak{h}}^{\beta}_{t,2}$. Then according to the Gibbs property
(Lemma 2.9), it suffices to bound

$Z^{-1}W(B,\mathfrak{h}^{\beta}_{t,2}),$ (B.3)

where

  * $\bullet$

$\hat{B}:[-L^{1/2}/2,L^{1/2}/2]\to\mathbb{R}$ is a rate $2$ Brownian bridge,
conditional on
$\hat{B}(-L^{1/2}/2)=\hat{\mathfrak{h}}^{\beta}_{t,1}(-L^{1/2}/2)$,
$\hat{B}(0)=L$,
$\hat{B}(L^{1/2}/2)=\hat{\mathfrak{h}}^{\beta}_{t,1}(L^{1/2}/2)$; and
$B:[-t^{2/3}L^{1/2}/2,t^{2/3}L^{1/2}/2]\to\mathbb{R}$ satisfies that
$\hat{B}(x)=t^{-1/3}B(t^{2/3}x)$;

  * $\bullet$

$W(B,\mathfrak{h}^{\beta}_{t,2})<1$ is the weight defined through (2.6), in
the interval $[-t^{2/3}L^{1/2}/2,t^{2/3}L^{1/2}/2]$;

  * $\bullet$

$Z=\mathbb{E}[W(B,\mathfrak{h}^{\beta}_{t,2})\mid\mathcal{F}]$ is a
renormalization constant.

Assuming $\mathcal{E}_{0}\cap\mathcal{E}_{1}\cap\mathcal{E}_{2}$, we have
$\mathbb{P}(\hat{B}(x)\geq 0.2L,\forall
x\in[-L^{1/2}/2,L^{1/2}/2]\mid\mathcal{F})>1-C\exp(-cL^{2})$; and whenever
$\inf_{x\in[-L^{1/2}/2,L^{1/2}/2]}\hat{B}(x)\geq 0.2L$ and also under
$\mathcal{E}_{1}$, we have

$\mathfrak{h}^{\beta}_{t,2}(t^{2/3}x)-B(t^{2/3}x)>t^{1/3}\cdot
0.1L,\quad\forall x\in[-L^{1/2}/2,L^{1/2}/2].$

Therefore, from (2.6) we have

$W(B,\mathfrak{h}^{\beta}_{t,2})>1-2t^{2/3}L^{1/2}\exp(-t^{1/3}\cdot
0.1L)>1-C\exp(-ct^{1/3}L).$

Thus we have $Z>1-C\exp(-ct^{1/3}L)$, and (B.3) is $1+O(\exp(-ct^{1/3}L))$
under $\mathcal{E}_{0}\cap\mathcal{E}_{1}\cap\mathcal{E}_{2}$. This combined
with the estimates on $\mathbb{P}(\mathcal{E}_{1}\mid\mathcal{E}_{0})$ and
$\mathbb{P}(\mathcal{E}_{2}\mid\mathcal{E}_{0})$ leads to the conclusion. ∎

We next prove Lemma 2.20.

###### Proof of Lemma 2.20.

For the first bound, by Theorem 2.17 we have that

$\mathbb{P}\left(|\hat{\mathfrak{h}}^{\beta}_{t,1}(\pm L^{1/2}/2)|>ML^{1/4}/5\
\Big{|}\
\hat{\mathfrak{h}}^{\beta}_{t,1}(0)\in(L,L+\,\mathrm{d}L)\right)<\exp(-cM^{2}).$

Then the bound follows from Lemma 2.19 and standard Brownian bridge estimates.

For the second bound, we use a strategy similar to the proof of Lemma 2.18.
Namely, the LHS is bounded by

$\displaystyle\sup_{L^{\prime}\in(L,L+M\sigma_{I}/2)}\mathbb{P}\left(\sup_{x\in
I}\left|\hat{\mathfrak{h}}^{\beta}_{t,1}(x)-(L-2L^{1/2}|x|)\right|>M\sigma_{I}\
\Big{|}\
\hat{\mathfrak{h}}^{\beta}_{t,1}(0)\in(L^{\prime},L^{\prime}+\,\mathrm{d}L^{\prime})\right)$
$\displaystyle\qquad+\mathbb{P}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L+M\sigma_{I}/2\
\Big{|}\ \hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L\right).$

Then by the first bound and Theorem 4.1, the second bound follows. ∎

##  Appendix C Tail and tent estimates for small time

In this appendix we explain how to adapt the proofs in [GH22] to get Theorem
2.13 and Theorem 2.17, using inputs from [DG23]. Theorems 2.13 and 2.17 are
proven in [GH22] for $t>t_{0}$ for any $t_{0}$ fixed, so here we prove it for
$0<t<t_{0}$ for a particular $t_{0}$ coming from the new inputs from [DG23].

Written with our scaling, [DG23, Theorem 1.4] asserts that, for any
$\varepsilon>0$, there exist $t_{0}$, $c$, and $s_{0}$, all depending on
$\varepsilon$, such that, for $0<t<t_{0}$ and $M>M_{0}t^{-1/12}$,

$\displaystyle\mathbb{P}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>M+t^{-1/3}\log
t^{-1}\right)\leq\exp\left(-c\frac{M^{2}t^{1/6}}{\sqrt{1+Mt^{1/3-\varepsilon/2}}}\right).$

If we additionally assume that $M>M_{0}t^{-1/3-\varepsilon}$ and $t<1$, so
that $M^{1/2}\geq t^{-1/6-\varepsilon/2}\geq t^{-\varepsilon/2}$, the previous
display implies that

$\displaystyle\mathbb{P}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>2M\right)\leq\mathbb{P}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>M+t^{-1/3}\log
t^{-1}\right)\leq\exp\left(-cM^{3/2}t^{\varepsilon/2}\right)\leq\exp(-cM).$

For control on the lower tail, as recorded in Theorem 2.15, [DG23, Theorem
1.7] and a similar calculation shows that, if $M>t^{-1/6}$,

$\displaystyle\mathbb{P}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)<-M\right)\leq\exp(-cM^{2}t^{1/6})\leq\exp(-cM);$
(C.1)

and one can also instead assume $M>t^{-1/12-\varepsilon}$ and obtain a weaker
bound of the form $\exp(-cM^{\alpha})$ for some $\alpha=\alpha(\varepsilon)$.

To apply the arguments of [GH22], we also need an a priori lower bound on
$\mathbb{P}(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>M)$. For this it is easy to
check that the proof of [GH22, Lemma 5.4] applies verbatim if we assume
$M>\smash{t^{-1/6}}$ (or $M>\smash{t^{-1/12-\varepsilon}}$, as above) and use
(C.1) in place of lower tail tightness (over $t\geq t_{0}$) of
$\hat{\mathfrak{h}}^{\beta}_{t,1}(0)$. This will then yield, for $t>0$ and
$M>t^{-1/6}$,

$\displaystyle\mathbb{P}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>M\right)\geq\exp(-5M^{3/2}).$

With these estimates in place, we satisfy Assumptions (i)–(iv) from [GH22,
Section 2.2] (more precisely, the above verifies Assumption (iv) on tail
bounds, as the other assumptions are qualitative and hold for any $t>0$). In
particular, [GH22, Theorems 4.1, 5.1, and 5.3] will apply as they hold only
assuming these assumptions, modulo controlling the partition function arising
from the Gibbs property of $\hat{\mathfrak{h}}^{\beta}_{t}$, and we explain
this point next.

In [GH22], the estimates lower bounding the partition function are captured in
Lemma 3.11, Corollary 3.12, Proposition 3.14, and Corollary 3.15. The latter
two are already stated for $t>0$. Lemma 3.11 says that the partition function
$Z_{t}$ of a single curve with respect to a lower boundary curve $p$ on an
interval $[z_{1},z_{2}]$ is lower bounded by
$\exp(-2t^{2/3}e^{-t^{1/6}})\int_{z_{1}}^{z_{2}}\exp(-t^{1/3}g(u))\,\mathrm{d}u)\cdot\mathbb{P}(B(u)>p(u)+g(u)+t^{-1/6},\forall
u\in[z_{1},z_{2}])$ for any non-negative function $g$, and a quick inspection
of the proof shows that a minor modification yields a lower bound of
$\exp(-2t^{2/3})\int_{z_{1}}^{z_{2}}\exp(-t^{1/3}g(u))\,\mathrm{d}u)\cdot\mathbb{P}(B(u)>p(u)+g(u),\forall
u\in[z_{1},z_{2}])$, which is better for small $t$. Carrying this change to
Corollary 3.12 and then making use of these estimates in the proofs of [GH22,
Theorems 4.1, and 5.1] will then yield Theorems 2.13 and 2.17.

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