dakling.sty

\usepackage{xparse}
\usepackage{yfonts}
\usepackage{eucal}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsfonts}
% \usepackage{amsthm}
\usepackage{csquotes}
\RequirePackage{xcolor}
% \usepackage[noabbrev,capitalise]{cleveref}
\usepackage[capitalise]{cleveref}
\usepackage{xifthen}

\makeatletter
\newcommand*{\MyIfEmptyTF}[1]{%
  \if\relax\detokenize{#1}\relax%
    \expandafter\@firstoftwo%
  \else%
    \expandafter\@secondoftwo%
  \fi
  }

\newcommand*\foreachletter[2]{%
        \begingroup
        \let\templettercommand#1%
        \let\tempspacecommand#2%
        \catcode
        \foreachlettergo
}
\def\foreachlettergo#1{%
        \testletter#1\relax
        \endgroup
}
\def\testletter#1#2\relax{%
        \if#1\otherspace
                \tempspacecommand
        \else
                \templettercommand{#1}%
        \fi
        \ifx\relax#2\relax
                \let\next\relax
        \else
                \let\next\testletter
        \fi
        \next#2\relax
}\catcode`\ 12
\def\otherspace{ }%
\catcode`\ 10
\makeatother

% Default settings for all available options
\newtoggle{EinsteinNotationDiff} % affects second derivatives
\newtoggle{ShortHigherMoments} % abbreviate higher one-point velocity moments
\newtoggle{ShowArgsTwoPoint} % show arguments eg in H_ij(x,y) 
\newtoggle{GroupExpSimplified} % use linear group parameter for scaling symmetries, not exp(a)
\newtoggle{OnePointExponent} % show ^0 for H and R one-point
\newtoggle{EinsteinSumConvention} % Hide sum signs if Einstein summation convention is active

%% general settings
%allow double superscripts
% \catcode`\^ = 13 \def^#1{\sp{#1}{}}
% \catcode`\_ = 13 \def_#1{\sb{#1}{}}

\newcommand\posscite[1]{\citeauthor{#1}'s (\citeyear{#1})}

% use with align*-environment
\newcommand*\donumber{\addtocounter{equation}{1}\tag{\theequation}}
%% low-level macros
% definition of a good looking bar
\newcommand*{\overbar} [ 1 ] {\mkern 1.5mu\overline{\mkern-1.5mu{#1}\mkern-1.5mu}\mkern 1.5mu}
% \newcommand*{\overbar} [ 1 ] {\widebar{#1}}

%% semantic macros
% real numbers
\newcommand*{\real} [1][] {
    % \MyIfEmptyTF{#1}
        % {\mathbb{R} }
        {\mathbb{R}^{#1} }
        }
% symbols to denote entire equations
\newcommand*{\nc}{ \mathcal{N} }
\newcommand*{\ncm}{ \mean{\mathcal{N}} }
\newcommand*{\ncf}{ \tilde{\mathcal{N}} }

\newcommand*{\mc}{ \mathcal{M} }
\newcommand*{\mcm}{ \mean{\mathcal{M}} }
\newcommand*{\mcf}{ \tilde{\mathcal{M}} }

% generic vector/tensor
\providecommand*{\vv}[1]{\mathbf{#1}}
\renewcommand*{\vv}[1]{\mathbf{#1}}
% \renewcommand*{\vv}[1]{\boldsymbol #1}
\newcommand*{\xv}{ \vv{x} }
\newcommand*{\yv}{ \vv{y} }
\newcommand*{\uv}{ \vv{u} }
\newcommand*{\zv}{ \vv{z} }
% \newcommand*{\te}[1]{ \mathbf{#1} }
\newcommand*{\te}[1]{ \boldsymbol #1 }
\newcommand*{\xt}{ \te{x} }
\newcommand*{\yt}{ \te{y} }
\newcommand*{\transpose}[1]{ {#1}^{\intercal} }
% absolute value
\newcommand*{\abs}[1]{\left| #1 \right|}
% "order of" symbol
\newcommand*{\orderBase}{\textit{O}}
\newcommand*{\order} [2] {
    \MyIfEmptyTF{#2}
    {\orderBase(#1)}
    {\orderBase({#1}^{#2}) }
}
\newcommand*{\code} [ 1 ] {\texttt{#1}}
\newcommand*{\file} [ 1 ] {\texttt{#1}}
% evaluate (eg derivative) at position
\newcommand*{\atpos} [2] { \bigg|_{{#1} = {#2}} } % use evalfun instead!
\newcommand*{\evalfunI} [3] {\left.\kern-\nulldelimiterspace {#1} \right|_{{#2} = {#3}}}
\newcommand*{\evalfun} [3] {\left. #1 \right|_{{#2} = {#3}}}
% transformed (symmetries)
\newcommand*{\tra}[2][]{\traAltEq[#1]{#2}}
\newcommand*{\traAnco}[2][]{#2^{#1} \rightarrow}
\newcommand*{\traAlt}[2][]{{#2^{* #1}}}
\newcommand*{\traAltEq}[2][]{{#2^{* #1} =\; }}

% nondimensional variables
\newcommand*{\diml} [1] {{#1}^{+}}

\newcommand*{\einsteinSum} [2][] {
    \iftoggle{EinsteinSumConvention}
        {\MyIfEmptyTF{#1}
            {}
            \else
            {#1}
            \fi}
        {\MyIfEmptyTF{#1}
            {\sum_{{#2}=1}^{3}}
            {\sum_{{#2}=1}^{3} \left( {#1} \right)}}}
% representation of similarity variables
\newcommand*{\simvareta}{\eta}
\newcommand*{\simvarxi}{\xi}
\newcommand*{\simvar} [1] {\widetilde{#1}}
% averaging
\ExplSyntaxOn
\newcommand*{\mean} [1] {
    \tl_set:Nn \l_tmpa_tl { #1 } % store argument in variable \l_tmpa_tl
    \int_compare:nTF { \tl_count:N \l_tmpa_tl = 1 } % check if the argument only contains a single value
        { \bar{#1} } % if there is only one letter, use the bar macro
        { \overline{#1}} % otherwise use the longer overline
    }
% derivatives
\newcommand*{\diffGeneric}[4][]{
    \MyIfEmptyTF{#1}
        { \cfrac{#4 #2}{#4 #3} }
        { \int_compare:nTF{ #1=2 } % second derivatives in Einstein notation
            {\iftoggle{EinsteinNotationDiff} % check if option is enabled
                {\cfrac{#4^{#1} #2}{#4 #3 #4 #3} }
                {\cfrac{#4^{#1} #2}{#4 #3^{#1}} }
            } 
            {{\cfrac{#4^{#1} #2}{#4 #3 ^{#1}}}} % higher derivatives, Einstein notation won't make sense in general
        }
}
\ExplSyntaxOff
% total derivative
\newcommand*{\dd} [3][] { \diffGeneric[#1]{#2}{#3}{d} }
% partial derivative
\newcommand*{\pd} [3][] { \diffGeneric[#1]{#2}{#3}{\partial} }
% material derivative
\newcommand*{\md} [3][] { \diffGeneric[#1]{#2}{#3}{D} }
\newcommand*{\mda} [3][] { \diffGeneric[#1]{#2}{#3}{\hat{D}} }
% total derivative (jet notation)
\newcommand*{\td} [3][] { \diffGeneric[#1]{#2}{#3}{\mathcal{D}} }
% derivatives without frac
\newcommand*{\diffInline} [3] { {#3 #1}/{#3 #2} }
\newcommand*{\did} [2] { \diffInline{#1}{#2}{d} }
\newcommand*{\pdi} [2] { \diffInline{#1}{#2}{\partial} }
\newcommand*{\mdi} [2] { \diffInline{#1}{#2}{D} }
\newcommand*{\tdi} [2] { \diffInline{#1}{#2}{\mathcal{D}} }
\newcommand*{\mdai} [2] { \diffInline{#1}{#2}{\hat{D}} }
\newcommand*{\vdi} [2] { \diffInline{#1}{#2}{\delta} }

%variational derivative
\newcommand*{\vd} [3][] { \diffGeneric[#1]{#2}{#3}{\delta} }
% jet derivative
% \newcommand*{\jd} [2] { \underset{#2}{#1} }
% \newcommand*{\jd} [2] { {#1}_{{\!,{#2}}} }
\newcommand*{\jd} [2] { {#1}{}_{{\!,{#2}}} }

% for a long first arg it is nicer to write d/dx(z+longArg)
\newcommand*{\diffLongGeneric} [4][] {\diffGeneric[#1]{}{#3}{#4}{#2}}
\newcommand*{\ddl} [3][] { \diffLongGeneric[#1]{#2}{#3}{d} }
\newcommand*{\pdl} [3][] { \diffLongGeneric[#1]{#2}{#3}{\partial} }
\newcommand*{\mdl} [3][] { \diffLongGeneric[#1]{#2}{#3}{D} }
\newcommand*{\tdl} [3][] { \diffLongGeneric[#1]{#2}{#3}{\mathcal{D}} }

\newcommand*{\diffSecond} [4] {
    \cfrac{{#1} ^2 {#2}}{{#1} {#3} {#1} {#4}}
}
% total derivative
\newcommand*{\ddSecond} [3] { \diffSecond{d}{#1}{#2}{#3} }
% partial derivative
\newcommand*{\pdSecond} [3] { \diffSecond{\partial}{#1}{#2}{#3} }
% material derivative
\newcommand*{\mdSecond} [3] { \diffSecond{D}{#1}{#2}{#3} }
\newcommand*{\tdSecond} [3] { \diffSecond{\mathcal{D}}{#1}{#2}{#3} }

\newcommand*{\xc}[1]{x_{#1}}
\newcommand*{\zc}[1]{z_{#1}}
% \newcommand*{\nondim}[1]{{}^+ {#1}}
\newcommand*{\xnd}[1]{x_{#1^+}}
\newcommand*{\tc}{t}
\newcommand*{\deltat}{\Delta t}

% velocity moments
\newcommand*{\ui}[1]{U_{#1}} % unaveraged velocity
\newcommand*{\uif}[1]{u_{#1}} % unaveraged fluctuating velocity

\newcommand*{\umv}  {
\bar{\vv{U}}
}
\newcommand*{\uav}  {
\hat{\vv{U}}
}
\newcommand*{\ufv}  {
\mean{\vv{u} \otimes \vv{u}}
}

\newcommand*{\ums}[1]  {
  \mathfrak{{U}}_{#1}
}

\newcommand*{\uflucthigherbase}{R}
\newcommand*{\uinsthigherbase}{H}

\newcommand*{\ufFirst} [1] { u_{#1} }
\newcommand*{\ufHigherShort} [1] {%
        \iftoggle{OnePointExponent}
        {\uflucthigherbase^{(0)}_{#1}}
        {\uflucthigherbase_{#1}}
        }
\newcommand*{\uftHigherShort} [1] {%
        \uflucthigherbase_{#1}
        }

\newcommand*{\ufHigherLong} [1] {%
    \mean{\foreachletter{\uif}{}{#1}}
}

\newcommand*{\ufHigher} [1] {%
        \iftoggle{ShortHigherMoments}
        {\ufHigherShort{#1}} 
        {\ufHigherLong{#1}}
        }

\newcommand*{\uftHigher} [1] {%
        \iftoggle{ShortHigherMoments}
        {\uftHigherShort{#1}} 
        {\ufHigherLong{#1}}
        }

\newcommand*{\umFirst} [1] { \mean{U}_{#1} }
\newcommand*{\umHigherShort} [1] {%
        \iftoggle{OnePointExponent}
        {\uinsthigherbase^{(0)}_{#1}}
        {\uinsthigherbase_{#1}}
        }

\newcommand*{\umHigherLong} [1] {%
    \mean{\foreachletter{\ui}{}{#1}}
}

\newcommand*{\umtHigherShort} [1] {%
        \uinsthigherbase_{#1}
        }

\newcommand*{\umtHigherLong} [1] {%
    \mean{\foreachletter{\ui}{}{#1}}
}

\newcommand*{\umHigher} [1] {%
        \iftoggle{ShortHigherMoments}
        {\umHigherShort{#1}} 
        {\umHigherLong{#1}}
        }

\newcommand*{\umtHigher} [1] {%
        \iftoggle{ShortHigherMoments}
        {\umtHigherShort{#1}} 
        {\umHigherLong{#1}}
        }

\ExplSyntaxOn
\newcommand*{\uf} [1] {
    \tl_set:Nn \l_tmpa_tl { #1 } % store argument in variable \l_tmpa_tl
    \int_compare:nTF { \tl_count:N \l_tmpa_tl = 1 } % check if the argument only contains a single value
        {\ufFirst{#1}} %
        {\ufHigher{#1}} %
    }
\newcommand*{\uft} [1] {
    \tl_set:Nn \l_tmpa_tl { #1 } % store argument in variable \l_tmpa_tl
    \int_compare:nTF { \tl_count:N \l_tmpa_tl = 1 } % check if the argument only contains a single value
        {\ufFirst{#1}} %
        {\uftHigher{#1}} %
    }
\newcommand*{\um} [1] {
    \tl_set:Nn \l_tmpa_tl { #1 } % store argument in variable \l_tmpa_tl
    \int_compare:nTF { \tl_count:N \l_tmpa_tl = 1 } % check if the argument only contains a single value
        {\umFirst{#1}} %
        {\umHigher{#1}} %
    }
\newcommand*{\umt} [1] {
    \tl_set:Nn \l_tmpa_tl { #1 } % store argument in variable \l_tmpa_tl
    \int_compare:nTF { \tl_count:N \l_tmpa_tl = 1 } % check if the argument only contains a single value
        {\umFirst{#1}} %
        {\umtHigher{#1}} %
    }
\ExplSyntaxOff
\renewcommand*{\pm}{\mean{P}}
\newcommand*{\pin}{P}
\newcommand*{\pf}{p}
\ExplSyntaxOn
\newcommand*{\umTensor} [1] {
    \int_compare:nTF { #1 = 1 } % 
    { \vv{\um} }
    { \vv{H^0} }
}
\newcommand*{\ufTensor} [1] {
    \int_compare:nTF { #1 = 1 } % 
    { \vv{\uf} }
    { \vv{R^0} }
}
\newcommand*{\umTensorTwoPt} [1] {
    \int_compare:nTF { #1 = 1 } % 
    { \vv{\um} }
    { \vv{H} }
}
\newcommand*{\ufTensorTwoPt} [1] {
    \int_compare:nTF { #1 = 1 } % 
    { \vv{\uf} }
    { \vv{R} }
}
\ExplSyntaxOff

\newcommand*{\ufmodeled} [1] {
  % \widetilde{\uf{#1}}
  \tilde{R}_{#1}
}

\newcommand*{\ufinvmodeled} [1] {
  \hat{R}_{#1}
}

\newcommand*{\pum} [1] {
        {\iftoggle{OnePointExponent}
        {\mean{PU_{#1}}^{(0)}}
        {\mean{PU_{#1}}}
        }
        % {\mean{PU_{#1}}(\vv{#2},\vv{#3})}
}
\newcommand*{\upm} [1] {
        {\iftoggle{OnePointExponent}
        {\mean{U_{#1}P}^{(0)}}
        {\mean{U_{#1}P}}
        }
        % {\mean{U_{#1}P}(\vv{#2},\vv{#3})}
}
\newcommand*{\umpx} [2] {
        {\iftoggle{OnePointExponent}
          {\mean{U_{#1} \jd{P}{x_{#2}}}^{(0)}}
          {\mean{U_{#1} \jd{P}{x_{#2}}}}
          }
        % {\mean{U_{#1}P}(\vv{#2},\vv{#3})}
}
\newcommand*{\puf} [1] {
        {\iftoggle{OnePointExponent}
        {\mean{pu_{#1}}^{(0)}}
        {\mean{pu_{#1}}}
        }
        % {\mean{pu_{#1}}(\vv{#2},\vv{#3})}
}
\newcommand*{\upf} [1] {
        {\iftoggle{OnePointExponent}
        {\mean{u_{#1}p}^{(0)}}
        {\mean{u_{#1}p}}
        }
        % {\mean{u_{#1}p}(\vv{#2},\vv{#3})}
}
% two-point correlations
% instantaneous approach
% \newcommand*{\umtWithoutArgsShort} [8] {
%     \MyIfEmptyTF{#1}
%             { \um }
%             {\MyIfEmptyTF{#2}
%                 % {\um{#1} }
%                 {\um{#1}(\vv{#5}) }% always show argument for single variable
%                 { \MyIfEmptyTF{#3} 
%                     {H_{#1 #2} }
%                     { \MyIfEmptyTF{#4}
%                         {H_{#1 #2 #3} }
%                     {H_{#1 #2 #3 #4} }
%                 }
%             }
%         }
%      }
% \newcommand*{\umtWithArgsShort} [8] {
%         \MyIfEmptyTF{#1}
%             { \um(\vv{#5}) }
%             {\MyIfEmptyTF{#2}
%                 {\um{#1}(\vv{#5}) }
%                 { \MyIfEmptyTF{#3} 
%                     {H_{#1 #2}(\vv{#5},\vv{#6}) }
%                     { \MyIfEmptyTF{#4}
%                         {H_{#1 #2 #3}(\vv{#5}, \vv{#6}, \vv{#7}) }
%                         {H_{#1 #2 #3 #4}(\vv{#5}, \vv{#6}, \vv{#7}, \vv{#8}) }
%                 }
%             }
%         }
%  }
% \newcommand*{\umtShort} [8] {
%     \MyIfEmptyTF{#5}
%     { \umtWithoutArgsShort{#1}{#2}{#3}{#4}[#5][#6][#7][#8] }
%     { \iftoggle{ShowArgsTwoPoint}
%         { \umtWithArgsShort{#1}{#2}{#3}{#4}[#5][#6][#7][#8] }
%         { \umtWithoutArgsShort{#1}{#2}{#3}{#4}[#5][#6][#7][#8] }
%     }
% }
% \newcommand*{\umtWithoutArgsLong} [8] {
%     \MyIfEmptyTF{#1}
%             { \um }
%             {\MyIfEmptyTF{#2}
%                 {\um{#1} }
%                 {\um{#1}(\vv{#5}) }% always show argument for single variable
%                 { \MyIfEmptyTF{#3} 
%                     { \mean{ U_{#1} U_{#2} } }
%                     { \MyIfEmptyTF{#4}
%                         { \mean{U_{#1} U_{#2} U_{#3}} }
%                         { \mean{U_{#1} U_{#2} U_{#3} U_{#4}} }
%                 }
%             }
%         }
%      }
% \newcommand*{\umtWithArgsLong} [8] {
%         \MyIfEmptyTF{#1}
%             { \um(\vv{#5}) }
%             {\MyIfEmptyTF{#2}
%                 {\um{#1}(\vv{#5}) }
%                 { \MyIfEmptyTF{#3} 
%                     { \mean{ U_{#1}(\vv{#5}) U_{#2}(\vv{#6})} }
%                     { \MyIfEmptyTF{#4}
%                         { \mean{ U_{#1}(\vv{#5}) U_{#2}(\vv{#6}) U_{#3}(\vv{#7})} }
%                         { \mean{ U_{#1}(\vv{#5}) U_{#2}(\vv{#6}) U_{#3}(\vv{#7}) U_{#4}(\vv{#8})} }
%                 }
%             }
%         }
%  }
% \newcommand*{\umtLong} [8] {
%     \MyIfEmptyTF{#5}
%     { \umtWithoutArgsLong{#1}{#2}{#3}{#4}[#5][#6][#7][#8] }
%     { \iftoggle{ShowArgsTwoPoint}
%         { \umtWithArgsLong{#1}{#2}{#3}{#4}[#5][#6][#7][#8] }
%         { \umtWithoutArgsLong{#1}{#2}{#3}{#4}[#5][#6][#7][#8] }
%     }
% }
%  \newcommand*{\umt} [8] {
%      \iftoggle{ShortHigherMoments}
%      {\umtShort{#1}{#2}{#3}{#4}[#5][#6][#7][#8]}
%      {\umtLong{#1}{#2}{#3}{#4}[#5][#6][#7][#8]}
%  }
% % fluctuation approach
% \newcommand*{\uftWithoutArgsShort} [4] {
%     \MyIfEmptyTF{#1}
%             { \uf }
%             {\MyIfEmptyTF{#2}
%                 {\uf{#1} }
%                 { \MyIfEmptyTF{#3} 
%                     {R_{#1 #2} }
%                     { \MyIfEmptyTF{#4}
%                         {R_{#1 #2 #3} }
%                     {R_{#1 #2 #3 #4} }
%                 }
%             }
%         }
%      }
% \newcommand*{\uftWithArgsShort} [8] {
%         \MyIfEmptyTF{#1}
%             { \uf(\vv{#5}) }
%             {\MyIfEmptyTF{#2}
%                 {\uf{#1}(\vv{#5}) }
%                 { \MyIfEmptyTF{#3} 
%                     {R_{#1 #2}(\vv{#5},\vv{#6}) }
%                     { \MyIfEmptyTF{#4}
%                         {R_{#1 #2 #3}(\vv{#5}, \vv{#6}, \vv{#7}) }
%                         {R_{#1 #2 #3 #4}(\vv{#5}, \vv{#6}, \vv{#7}, \vv{#8}) }
%                 }
%             }
%         }
%  }
% \newcommand*{\uftShort} [8] {
%     \MyIfEmptyTF{#5}
%     { \uftWithoutArgsShort{#1}{#2}{#3}{#4} }
%     { \iftoggle{ShowArgsTwoPoint}
%         { \uftWithArgsShort{#1}{#2}{#3}{#4}[#5][#6][#7][#8] }
%         { \uftWithoutArgsShort{#1}{#2}{#3}{#4} }
%     }
% }
% \newcommand*{\uftWithoutArgsLong} [4] {
%     \MyIfEmptyTF{#1}
%             { \uf }
%             {\MyIfEmptyTF{#2}
%                 {\uf{#1} }
%                 { \MyIfEmptyTF{#3} 
%                     { \mean{ u_{#1} u_{#2} } }
%                     { \MyIfEmptyTF{#4}
%                         { \mean{U_{#1} u_{#2} u_{#3}} }
%                         { \mean{U_{#1} u_{#2} u_{#3} u_{#4}} }
%                 }
%             }
%         }
%      }
% \newcommand*{\uftWithArgsLong} [8] {
%         \MyIfEmptyTF{#1}
%             { \uf(\vv{#5}) }
%             {\MyIfEmptyTF{#2}
%                 {\uf{#1}(\vv{#5}) }
%                 { \MyIfEmptyTF{#3} 
%                     { \mean{ u_{#1}(\vv{#5}) u_{#2}(\vv{#6})} }
%                     { \MyIfEmptyTF{#4}
%                         { \mean{ u_{#1}(\vv{#5}) u_{#2}(\vv{#6}) u_{#3}(\vv{#7})} }
%                         { \mean{ u_{#1}(\vv{#5}) u_{#2}(\vv{#6}) u_{#3}(\vv{#7}) u_{#4}(\vv{#8})} }
%                 }
%             }
%         }
%  }
% \newcommand*{\uftLong} [8] {
%     \MyIfEmptyTF{#5}
%     { \uftWithoutArgsLong{#1}{#2}{#3}{#4} }
%     { \iftoggle{ShowArgsTwoPoint}
%         { \uftWithArgsLong{#1}{#2}{#3}{#4}[#5][#6][#7][#8] }
%         { \uftWithoutArgsLong{#1}{#2}{#3}{#4} }
%     }
% }
%  \newcommand*{\uft} [8] {
%      \iftoggle{ShortHigherMoments}
%      {\uftShort{#1}{#2}{#3}{#4}[#5][#6][#7][#8]}
%      {\uftLong{#1}{#2}{#3}{#4}[#5][#6][#7][#8]}
%  }
\newcommand*{\puft} [2][] {
    \MyIfEmptyTF{#1}
        {\mean{pu_{#2}}}
        {\mean{pu_{#2}}}
        % {\mean{pu_{#1}}(\vv{#2},\vv{#3})}
}
\newcommand*{\upft} [2][] {
    \MyIfEmptyTF{#1}
        {\mean{u_{#2}p}}
        {\mean{u_{#2}p}}
        % {\mean{u_{#1}p}(\vv{#2},\vv{#3})}
}
\newcommand*{\pumt} [2][] {
    \MyIfEmptyTF{#1}
        {\mean{PU_{#2}}}
        {\mean{PU_{#2}}}
        % {\mean{PU_{#1}}(\vv{#2},\vv{#3})}
}
\newcommand*{\upmt} [2][] {
    \MyIfEmptyTF{#1}
        {\mean{U_{#2}P}}
        {\mean{U_{#2}P}}
        % {\mean{U_{#1}P}(\vv{#2},\vv{#3})}
}
%LES
% \newcommand*{\ufilter} [4] {
%     \MyIfEmptyTF{#1}
%         { \tilde{U} }
%         {\MyIfEmptyTF{#2}
%             {\tilde{U}_{#1}}
%             { \MyIfEmptyTF{#3} 
%                 {\tau_{#1 #2}}
%                 { \MyIfEmptyTF{#4}
%                     {\tau_{#1 #2 #3}}
%                 {H_{#1 #2 #3 #4}}
%             }
%         }
%     }
% }
\newcommand*{\pfilter}{\tilde{P}}

% conditional means
\newcommand*{\umturb}[1]{\mean{U}_{#1}^{\text{turb}}}
\newcommand*{\umlam}[1]{\mean{U}_{#1}^{\text{lam}}}


% turbulent viscosity
% \newcommand*{\nut}  {\nu_{\text{turb}}}
\newcommand*{\nut}  {\nu_{t}}
\newcommand*{\tke}{k}
\newcommand*{\tdr}{\varepsilon}
\newcommand*{\stdr}{\omega}
% artificial velocity
\newcommand*{\uaBase}{\hat{U}}
\newcommand*{\ua} [1] {
    % \MyIfEmptyTF{#1}
    % {\hat{U}} 
    \uaBase_{#1}
} 
\newcommand*{\pa}{ \hat{P} }
\newcommand*{\ufabase}{\hat{R}}
\newcommand*{\ufa}[1]{\ufabase_{#1}}
\newcommand*{\umh}[1]{\uinsthigherbase_{#1}}
\newcommand*{\ka}  { \hat{k} }
\newcommand*{\stdra}  { \hat{\omega} }
\newcommand*{\tdra}  { \hat{\varepsilon} }
\providecommand*{\ualpha}{}
\renewcommand*{\ualpha} [1][] {
    \MyIfEmptyTF{#1}
        {v} 
        {v_{#1}} 
} 
% k- eps - model
\newcommand*{\kepsmodel}{{\(k\)-\(\varepsilon\)-model}}
\newcommand*{\Kepsmodel}{{\(k\)-\(\varepsilon\)-Model}}
\newcommand*{\keps}{{\(k\)-\(\varepsilon\)}}
% k- omega - model
\newcommand*{\komegamodel}{{\(k\)-\(\omega\)-model}}
\newcommand*{\Komegamodel}{{\(k\)-\(\omega\)-Model}}
\newcommand*{\komega}{{\(k\)-\(\omega\)}}
\newcommand*{\klimited}{{k^{\prime}}}
\newcommand*{\omegatrans}{\tilde{\omega}}
\newcommand*{\omegatranslimited}{\tilde{\omega}^{\prime}}
\newcommand*{\omegarealz}{\omega_{\text{min}}}
% \newcommand*{\omegarealz}{\omega_{\text{realizable}}}
\newcommand*{\omegaITOomegatrans}  {{\exp{(\tilde{\omega}^{\prime})}}}
%Symmetry generators
\newcommand*{\infGenBase}{X}
\newcommand*{\infGenEvo} [1][] {%
    \MyIfEmptyTF{#1}
    {\hat{\infGenBase}}
        {\hat{\infGenBase_{#1}}}
    }
\newcommand*{\infGen} [1][] {%
    \MyIfEmptyTF{#1}
    {\infGenBase}
        {\infGenBase_{#1}}
    }
    \newcommand*{\infGenT}  {\infGen[t]}
\newcommand*{\infGenX} [1][] {
    \MyIfEmptyTF{#1} 
    {\infGen[x_i]}
    {\infGen[x_{#1}]}
}
\newcommand*{\infGenGal} [1][] {
    \MyIfEmptyTF{#1} 
    {\infGen[\text{Gal}]}
    {\infGen[{\text{Gal}_{{#1}}}]}
}
\newcommand*{\infGenGalDiff} [1][] {
    \MyIfEmptyTF{#1} 
    {\tilde{\infGen}_{\text{Gal}}}
    {\tilde{\infGen}_{{\text{Gal}_{{#1}}}}}
}
\newcommand*{\infGenRot} [1][]
{\MyIfEmptyTF{#1}
    {\infGen[{\text{rot}_{\alpha}}]}
    {\infGen[{\text{rot}_{#1}}]}}

\newcommand*{\infGenP}  {\infGen[P]}
\newcommand*{\infGenPDiff}  {\tilde{\infGen}_[P]}
\newcommand*{\infGenScI}  {\infGen[\text{Sc},I]}
\newcommand*{\infGenScII}  {\infGen[\text{Sc},II]}
\newcommand*{\infGenScNs}  {\infGen[\text{Sc,ns}]}
\newcommand*{\infGenScStat}  {\infGen[\text{Sc,stat}]}
\newcommand*{\infGenTransStat} [1] {\infGen[{\text{Tr,stat,{#1}}}]}
\newcommand*{\infGenTransStatI} [1][] {%
    \MyIfEmptyTF{#1}
        {\infGen[\text{Tr,stat},1]}
        {\infGen[\text{Tr,stat},1, {#1} ]}}
\newcommand*{\infGenTransStatII} [1][] {%
    \MyIfEmptyTF{#1}
        {\infGen[\text{Tr,stat},2]}
        {\infGen[\text{Tr,stat},2, {#1} ]}}
\newcommand*{\infGenTransStatIII} [1][] {%
    \MyIfEmptyTF{#1}
        {\infGen[\text{Tr,stat},3]}
        {\infGen[\text{Tr,stat},3, {#1} ]}}
%Global form
\newcommand*{\gloSymmBase}{T}
\newcommand*{\gloSymm} [1][] {
    \MyIfEmptyTF{#1}
    {\gloSymmBase}
    {\gloSymmBase_{#1}}
}
\newcommand*\gloSymmT{\gloSymm[t]}
\newcommand*{\gloSymmX} [1][] {
    \MyIfEmptyTF{#1}
        {\gloSymm[x_i]}
        {\gloSymm[x_{#1}]}
}
\newcommand*{\gloSymmGal} [1][] {
    \MyIfEmptyTF{#1}
        {\gloSymm[\text{Gal}_{i}]}
        {\gloSymm[\text{Gal}_{#1}]}
}
\newcommand*{\gloSymmRot}  {\gloSymm[\text{rot}_{\alpha}]}
\newcommand*{\gloSymmP}  {\gloSymm[P]}
\newcommand*{\gloSymmScI}  {\gloSymm[\text{Sc},I]}
\newcommand*{\gloSymmScII}  {\gloSymm[\text{Sc},II]}
\newcommand*{\gloSymmScNs}  {\gloSymm[\text{Sc,ns}]}
\newcommand*{\gloSymmScStat}  {\gloSymm[\text{Sc,stat}]}
\newcommand*{\gloSymmTransStat} [1][] {\gloSymm[\text{Tr,stat,{#1}}]}
\newcommand*{\gloSymmTransStatI} [1][] {%
    \MyIfEmptyTF{#1}
        {\gloSymm[\text{Tr,stat},I]}
        {\gloSymm[\text{Tr,stat},I, {#1} ]}}
\newcommand*{\gloSymmTransStatII} [1][] {%
    \MyIfEmptyTF{#1}
        {\gloSymm[\text{Tr,stat},II]}
        {\gloSymm[\text{Tr,stat},II, {#1} ]}}

\newcommand*{\lagX} [1] {\xi_{#1}}
% group parameter
\newcommand*{\grpBase}{a}
\newcommand*{\grp}[1][]{
    \MyIfEmptyTF{#1}
    {\grpBase}
    {\grpBase_{#1}}
}
\newcommand*{\grpT}  {\grp[T]}
\newcommand*{\grpX} [1][] {
    \MyIfEmptyTF{#1}
        {\grp[x_i]}
        {\grp[x_{#1}]}
}
\newcommand*{\grpGal} [1][] {
    \MyIfEmptyTF{#1}
        {\grp[\text{Gal}_{i}]}
        {\grp[\text{Gal}_{#1}]}
}
\newcommand*{\grpRot} [1][] {
    \MyIfEmptyTF{#1}
        {\grp[\text{rot}_{\alpha}]}
        {\grp[\text{rot}_{#1}]}
}
\newcommand*{\grpP}  {\grp[P]}
\newcommand*{\grpScI}  {\grp[\text{Sc},I]}
\newcommand*{\grpScII}  {\grp[\text{Sc},II]}
\newcommand*{\grpScNs}  {\grp[\text{Sc,ns}]}
\newcommand*{\grpScStat}  {\grp[\text{Sc,stat}]}
\newcommand*{\grpTransStat} [1] {\grp[\text{Tr,stat,{#1}}]}
\newcommand*{\grpTransStatI} [1][] {%
    \MyIfEmptyTF{#1}
        {\grp[\text{Tr,stat},I]}
        {\grp[\text{Tr,stat},I, {#1} ]}}
\newcommand*{\grpTransStatII} [1][] {%
    \MyIfEmptyTF{#1}
        {\grp[\text{Tr,stat},II]}
        {\grp[\text{Tr,stat},II, {#1} ]}}
\newcommand*{\grpTransStatIII} [1][] {%
    \MyIfEmptyTF{#1}
        {\grp[\text{Tr,stat},III]}
        {\grp[\text{Tr,stat},III, {#1} ]}}

% \newcommand*{ \expgrp }[1][] {
%     \iftoggle{GroupExpSimplified}
%     {\grp}
%     {e^{{#1} \grp}}
% }
%for galileian group
\newcommand*{\ftGal} [1] {f_{\text{Gal}_{#1}} (t)}
\newcommand*{\ftpGal} [1] {f_{\text{Gal}_{#1}}^{\prime} (t)}
\newcommand*{\ftppGal} [1] {f_{\text{Gal}_{#1}}^{\prime\prime} (t)}
\newcommand*{\ftpppGal} [1] {f_{\text{Gal}_{#1}}^{\prime\prime\prime} (t)}
%for pressure translation
\newcommand*{\ftP}  {f_{P} (t)}
\newcommand*{\ftPa}  {f_{\hat{P}} (t)}
\newcommand*{\ftpP}  {f_{P}^{\prime} (t)}
\newcommand*{\ftppP}  {f_{P}^{\prime\prime} (t)}
% model for a term
% \newcommand*{ \model }[1]{\mathcal{M}_{#1}}

% integration constant
\newcommand*{\intconstbase}{c}
\newcommand*{\intconst}[2][]{%
\MyIfEmptyTF{#1}
  {\intconstbase_{#2}}
  {\intconstbase_{{#2}_{#1}}}
}

% jet stuff
% \newcommand*{ \ucenterline }{U_{s}}

% cite author's name in possesive form
% \newcommand*{ \posscite } [1] { \citeauthor }
\newcommand*{\crefrangeconjunction}{--}

\providecommand*{\utau}{u_\tau}
\renewcommand*{\utau}{u_\tau}
\newcommand*{\retau}{\text{Re}_\tau}
\newcommand*{\karman}{\kappa}
% model parameters
\newcommand*{\cmu}{C_{\mu}}
\newcommand*{\cmuomegabase}{C_{\omega}}
\newcommand*{\cmuomega}[1][]{
\MyIfEmptyTF{#1}
  {\cmuomegabase}
  {C_{\omega, #1}}}
\newcommand*{\cepsprod}{C_{\varepsilon,1}}
\newcommand*{\cepsdiss}{C_{\varepsilon,2}}
\newcommand*{\cepsdiff}{c_{\varepsilon}}
\newcommand*{\cepshatdiff}{\hat{c}_{\varepsilon}}
\newcommand*{\cuadiff}{\hat{c}_{R}}
\newcommand*{\cepshatprod}{\hat{C}_{\varepsilon,1}}
% \newcommand*{\cepshatprodinv}{\hat{C}_{\varepsilon,3}}
\newcommand*{\cepshatprodinvratio}{\gamma}
\newcommand*{\cepshatdiss}{\hat{C}_{\varepsilon,2}}
\newcommand*{\sigmak}{\sigma_{k}}
\newcommand*{\sigmaeps}{\sigma_{\varepsilon}}
\newcommand*{\sigmaepshat}{\hat{\sigma}_{\varepsilon}}
\newcommand*{\modmomvisc}{\iota}

\newcommand*{\ckomegaprod}{\alpha^*}
\newcommand*{\ckomegadiss}{\beta^*}
\newcommand*{\comegaprod}{\alpha}
\newcommand*{\comegadiss}{\beta}
\newcommand*{\comegahatprod}{\alpha^{**}}
% \newcommand*{\cepshatprodinv}{\hat{C}_{\varepsilon,3}}
\newcommand*{\comegahatprodinvratio}{\gamma^*}
\newcommand*{\comegahatdiss}{\beta^{**}}
\newcommand*{\sigmakomega}{\sigma^*}
\newcommand*{\sigmaomega}{\sigma}
\newcommand*{\sigmaomegahat}{\sigma^{**}}
% DG stuff
\newcommand*{\jump}[1]{(#1^{\text{in}} - #1^{\text{out}})}
\newcommand*{\avg}[1]{\frac{#1^{\text{in}} + #1^{\text{out}}}{2}}
\newcommand*{\avgl}[1]{\frac{1}{2}\left(#1^{\text{in}} + #1^{\text{out}}\right)}
\newcommand*{\jumptwo}[2]{(#1^{\text{in}} #2^{\text{in}} - #1^{\text{out}} #2^{\text{out}})}
\newcommand*{\avgtwo}[2]{\frac{#1^{\text{in}} #2^{\text{in}} + #1^{\text{out}} #2^{\text{out}}}{2}}
\newcommand*{\avgltwo}[2]{\frac{1}{2}\left(#1^{\text{in}} #2^{\text{in}} + #1^{\text{out}} #2^{\text{out}}\right)}

\newcommand*{\gridsize}{h}
\newcommand*{\source}{S}
\newcommand*{\sourcemom}[1][i]{\source_{\text{mom},{#1}}}
\newcommand*{\sourcemass}{\source_{\text{conti}}}
\newcommand*{\velgradsymm}{S}
\newcommand*{\velgradantisymm}{W}
\newcommand*{\velgradhst}{S_{12}}
\newcommand*{\velgradhsta}{\hat{S}_{12}}

\newcommand*{\ordinal}[2][th]{#2{\text{#1}}}
\newcommand*{\nth}{\ordinal{n}}
\newcommand*{\ith}{\ordinal{i}}
\newcommand*{\jth}{\ordinal{j}}
\newcommand*{\first}{\ordinal[st]{1}}
\newcommand*{\second}{\ordinal[nd]{2}}
\newcommand*{\third}{\ordinal[rd]{3}}
\newcommand*{\fourth}{\ordinal{4}}
\newcommand*{\fifth}{\ordinal{5}}

% \newcommand*{\expnumber}[2]{{#1}\mathrm{e}{#2}} % scientific notation
\newcommand*{\expnumber}[2][]{
  \MyIfEmptyTF{#1}
  {10^{#2}}
  {{#1} \cdot 10^{#2}}
} % scientific notation

\DeclareMathOperator{\sgn}{sgn}

\newcommand*{\cahn}{\text{Cn}}
\newcommand*{\deformpar}{D_{\text{def}}}

\newcommand*{\fortran}{FORTRAN}
\newcommand*{\clang}{C}
\newcommand*{\cpp}{C++}
\newcommand*{\csharp}{C\#}
\newcommand*{\dotnet}{.NET}
\newcommand*{\dotnetfive}{.NET5}

% \newcommand*{\test}{v}

\newcommand*{\comment}[1]{\textcolor{blue}{#1}}


\newcommand*{\norm}[1]{\abs{#1}}

\newcommand*{\utot}[1][]{\vv{U_{#1}}}
\newcommand*{\ptot}{P}
\newcommand*{\upert}[1][]{\vv{\tilde{u}_{#1}}}
\newcommand*{\ppert}{\tilde{p}}
\newcommand*{\ubase}[1][]{\bar{\vv{U}_{#1}}}
\newcommand*{\ubaseargs}[1]{\bar{\vv{U}}({#1})}
\newcommand*{\pbase}[1][]{\bar{P_{#1}}}
\newcommand*{\pbaseargs}[1]{\bar{P}({#1})}

\newcommand*{\uadj}[1][]{\vv{\tilde{v}_{#1}}}
\newcommand*{\padj}[1][]{\tilde{q}_{#1}}

\newcommand*{\lagrangian}{\mathcal{L}}
\newcommand*{\volint}[1]{\iiint\limits_V{#1}dV}
\newcommand*{\timeint}[1]{\int_0^T{#1}dt}


\newcommand*{\energy}[1][]{
  \MyIfEmptyTF{#1}
  {\mathcal{E}}
  {\mathcal{E}\left({ #1 }\right)}
}
\newcommand*{\flux}[1][]{
  \MyIfEmptyTF{#1}
  {\dot{\mathcal{E}}}
  {\dot{\mathcal{E}}\left({ #1 }\right)}
}
\newcommand*{\dissflux}[1][]{
  \MyIfEmptyTF{#1}
  {\dot{\mathcal{E}}_{\text{diss}}}
  {\dot{\mathcal{E}}_{\text{diss}}\left({ #1 }\right)}
}