Rrerku

I want to visualize a turbulent flow regime. Assumed we have a simple empty square that should be filled with a pattern.

Minimum Working Example (MWE):

documentclassstandalone
usepackagetikz
begindocument

begintikzpicture 
 draw (0, 0) rectangle (4, 4);
endtikzpicture
enddocument

I won't post a screenshot, because it is just a simple rectangle. :-)

How can I fill this rectangle with a pattern like this on the right side:

Or, for the advanced TeX-users: How to fill it with a pattern like this one (made of random arrows)?

Thanks to marmot: As you can see, the arrows never intersect - this seems to be a challenge.

This is an answer to the question

How can one draw some randomly curved arrows that do not intersect?

which is not to be confused with

How can I draw the velocity field of some fluid?

which may require a model, a solution of the Navier-Stokes equations or something of that sort. That is, forbidding intersections is a step in the right direction but does not yield a physical description. If you do have the parametrization a realistic turbulent velocity field, you can do much better.

documentclass[tikz,border=3.14mm]standalone
usetikzlibraryintersections,arrows.meta,bending 
newcounterrandarcs
begindocument
begintikzpicture
%draw[clip] (0,0) rectangle (4,4);
pgfmathsetseed21
foreach X in 1,...,50
pgfmathsetmacromyx-0.5+5*rnd
pgfmathsetmacromyy-0.5+5*rnd
pgfmathsetmacroangA360*rnd
pgfmathsetmacroradA0.3+0.3*rnd
pgfmathsetmacromyxpmyx+radA*cos(angA)
pgfmathsetmacromyypmyy+radA*sin(angA)
pgfmathsetmacroangBangA-75+150*rnd
pgfmathsetmacroradBradA-0.1+0.2*rnd
pgfmathsetmacromyxqmyxp+radB*cos(angB)
pgfmathsetmacromyyqmyyp+radB*sin(angB)
pgfmathsetmacroangCangB-45+90*rnd
pgfmathsetmacroradCradB-0.1+0.2*rnd
pgfmathsetmacromyxrmyxq+radB*cos(angC)
pgfmathsetmacromyyrmyyq+radB*sin(angC)
%typeoutangA,radA;angB,radB
path[-Latex,name path=test-arc] plot[smooth,tension=1] 
coordinates (myx,myy) (myxp,myyp) (myxq,myyq) (myxr,myyr) ;
defHasIntersection0
ifnumX>1
 foreach Y in 1,...,numbervaluerandarcs
 path[name intersections=of=Y-arc and test-arc,total=t,
 /utils/exec=ifnumt>0
 xdefHasIntersection1%typeoutintersects
 fi];
 
fi
ifnumHasIntersection=0
 stepcounterrandarcs
 draw[-Latex[bend]] 
 plot[smooth,tension=1] coordinates (myx,myy) (myxp,myyp)
 (myxq,myyq) (myxr,myyr);
 path[name path global=numbervaluerandarcs-arc]
 plot[smooth,tension=1] coordinates (myx,myy) (myxp,myyp)
 (myxq,myyq) (myxr,myyr) -- cycle;
fi
endtikzpicture
typeoutnumbervaluerandarcsspace arcsspace drawn.
enddocument